{"id":399,"date":"2021-02-04T01:59:24","date_gmt":"2021-02-04T01:59:24","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=399"},"modified":"2022-03-16T05:44:26","modified_gmt":"2022-03-16T05:44:26","slug":"extrema-and-critical-points","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/extrema-and-critical-points\/","title":{"raw":"Extrema and Critical Points","rendered":"Extrema and Critical Points"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define absolute extrema<\/li>\r\n \t<li>Define local extrema<\/li>\r\n \t<li>Explain how to find the critical points of a function over a closed interval<\/li>\r\n \t<li>Describe how to use critical points to locate absolute extrema over a closed interval<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165041932730\" class=\"bc-section section\">\r\n<h2>Absolute Extrema<\/h2>\r\n<p id=\"fs-id1165040644979\">Consider the function [latex]f(x)=x^2+1[\/latex] over the interval [latex](\u2212\\infty ,\\infty )[\/latex]. As [latex]x\\to \\pm \\infty[\/latex], [latex]f(x)\\to \\infty [\/latex]. Therefore, the function does not have a largest value. However, since [latex]x^2+1\\ge 1[\/latex] for all real numbers [latex]x[\/latex] and [latex]x^2+1=1[\/latex] when [latex]x=0[\/latex], the function has a smallest value, 1, when [latex]x=0[\/latex]. We say that 1 is the<strong> absolute minimum<\/strong> of [latex]f(x)=x^2+1[\/latex] and it occurs at [latex]x=0[\/latex]. We say that [latex]f(x)=x^2+1[\/latex] does not have an<strong> absolute maximum<\/strong> (see Figure 1).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210800\/CNX_Calc_Figure_04_03_001.jpg\" alt=\"The function f(x) = x2 + 1 is graphed, and its minimum of 1 is seen to be at x = 0.\" width=\"487\" height=\"271\" \/> Figure 1. The given function has an absolute minimum of 1 at [latex]x=0[\/latex]. The function does not have an absolute maximum.[\/caption]\r\n<div id=\"fs-id1165041992939\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165041930563\">Let [latex]f[\/latex] be a function defined over an interval [latex]I[\/latex] and let [latex]c\\in I[\/latex]. We say [latex]f[\/latex] has an <strong>absolute maximum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex] if [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex]. We say [latex]f[\/latex] has an <strong>absolute minimum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex] if [latex]f(c)\\le f(x)[\/latex] for all [latex]x\\in I[\/latex]. If [latex]f[\/latex] has an absolute maximum on [latex]I[\/latex] at [latex]c[\/latex] or an absolute minimum on [latex]I[\/latex] at [latex]c[\/latex], we say [latex]f[\/latex] has an<strong> absolute extremum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex].<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165041811976\">Before proceeding, let\u2019s note two important issues regarding this definition. First, the term <em>absolute<\/em> here does not refer to absolute value. An absolute extremum may be positive, negative, or zero. Second, if a function [latex]f[\/latex] has an absolute extremum over an interval [latex]I[\/latex] at [latex]c[\/latex], the absolute extremum is [latex]f(c)[\/latex]. The real number [latex]c[\/latex] is a point in the domain at which the absolute extremum occurs. For example, consider the function [latex]f(x)=\\frac{1}{(x^2+1)}[\/latex] over the interval [latex](\u2212\\infty ,\\infty )[\/latex]. Since<\/p>\r\n\r\n<div id=\"fs-id1165041761810\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(0)=1\\ge \\dfrac{1}{x^2+1}=f(x)[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165041952862\">for all real numbers [latex]x[\/latex], we say [latex]f[\/latex] has an absolute maximum over [latex](\u2212\\infty ,\\infty )[\/latex] at [latex]x=0[\/latex]. The absolute maximum is [latex]f(0)=1[\/latex]. It occurs at [latex]x=0[\/latex], as shown in Figure 2b.<\/p>\r\nSo remember: the maximum\/minimum = [latex]y[\/latex]; the location of the maximum\/minimum =\u00a0[latex]x[\/latex].\r\n<p id=\"fs-id1165041795038\">A function may have both an absolute maximum and an absolute minimum, just one extremum, or neither. Figure 2 shows several functions and some of the different possibilities regarding absolute extrema. However, the following theorem, called the<strong> Extreme Value Theorem<\/strong>, guarantees that a continuous function [latex]f[\/latex] over a closed, bounded interval [latex][a,b][\/latex] has both an absolute maximum and an absolute minimum.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"923\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210805\/CNX_Calc_Figure_04_03_010.jpg\" alt=\"This figure has six parts a, b, c, d, e, and f. In figure a, the line f(x) = x3 is shown, and it is noted that it has no absolute minimum and no absolute maximum. In figure b, the line f(x) = 1\/(x2 + 1) is shown, which is near 0 for most of its length and rises to a bump at (0, 1); it has no absolute minimum, but does have an absolute maximum of 1 at x = 0. In figure c, the line f(x) = cos x is shown, which has absolute minimums of \u22121 at \u00b1\u03c0, \u00b13\u03c0, \u2026 and absolute maximums of 1 at 0, \u00b12\u03c0, \u00b14\u03c0, \u2026. In figure d, the piecewise function f(x) = 2 \u2013 x2 for 0 \u2264 x &lt; 2 and x \u2013 3 for 2 \u2264 x \u2264 4 is shown, with absolute maximum of 2 at x = 0 and no absolute minimum. In figure e, the function f(x) = (x \u2013 2)2 is shown on [1, 4], which has absolute maximum of 4 at x = 4 and absolute minimum of 0 at x = 2. In figure f, the function f(x) = x\/(2 \u2212 x) is shown on [0, 2), with absolute minimum of 0 at x = 0 and no absolute maximum.\" width=\"923\" height=\"900\" \/> Figure 2. Graphs (a), (b), and (c) show several possibilities for absolute extrema for functions with a domain of [latex](\u2212\\infty ,\\infty )[\/latex]. Graphs (d), (e), and (f) show several possibilities for absolute extrema for functions with a domain that is a bounded interval.[\/caption]\r\n<div id=\"fs-id1165042118849\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Extreme Value Theorem<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165042274290\">If [latex]f[\/latex] is a continuous function over the closed, bounded interval [latex][a,b][\/latex], then there is a point in [latex][a,b][\/latex] at which [latex]f[\/latex] has an absolute maximum over [latex][a,b][\/latex] and there is a point in [latex][a,b][\/latex] at which [latex]f[\/latex] has an absolute minimum over [latex][a,b][\/latex].<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165040759275\">The proof of the extreme value theorem is beyond the scope of this text. Typically, it is proved in a course on real analysis. There are a couple of key points to note about the statement of this theorem. For the extreme value theorem to apply, the function must be continuous over a closed, bounded interval. If the interval [latex]I[\/latex] is open or the function has even one point of discontinuity, the function may not have an absolute maximum or absolute minimum over [latex]I[\/latex]. For example, consider the functions shown in Figure 2(d), (e), and (f). All three of these functions are defined over bounded intervals. However, the function in graph (e) is the only one that has both an absolute maximum and an absolute minimum over its domain. The extreme value theorem cannot be applied to the functions in graphs (d) and (f) because neither of these functions is continuous over a closed, bounded interval. Although the function in graph (d) is defined over the closed interval [latex][0,4][\/latex], the function is discontinuous at [latex]x=2[\/latex]. The function has an absolute maximum over [latex][0,4][\/latex] but does not have an absolute minimum. The function in graph (f) is continuous over the half-open interval [latex][0,2)[\/latex], but is not defined at [latex]x=2[\/latex], and therefore is not continuous over a closed, bounded interval. The function has an absolute minimum over [latex][0,2)[\/latex], but does not have an absolute maximum over [latex][0,2)[\/latex]. These two graphs illustrate why a function over a bounded interval may fail to have an absolute maximum and\/or absolute minimum.<\/p>\r\n<p id=\"fs-id1165041805532\">Before looking at how to find absolute extrema, let\u2019s examine the related concept of local extrema. This idea is useful in determining where absolute extrema occur.<\/p>\r\n\r\n<\/div>\r\n<h2>Local Extrema and Critical Points<\/h2>\r\n<p id=\"fs-id1165042120195\">Consider the function [latex]f[\/latex] shown in Figure 3. The graph can be described as two mountains with a valley in the middle. The absolute maximum value of the function occurs at the higher peak, at [latex]x=2[\/latex]. However, [latex]x=0[\/latex] is also a point of interest. Although [latex]f(0)[\/latex] is not the largest value of [latex]f[\/latex], the value [latex]f(0)[\/latex] is larger than [latex]f(x)[\/latex] for all [latex]x[\/latex] near 0. We say [latex]f[\/latex] has a local maximum at [latex]x=0[\/latex]. Similarly, the function [latex]f[\/latex] does not have an absolute minimum, but it does have a local minimum at [latex]x=1[\/latex] because [latex]f(1)[\/latex] is less than [latex]f(x)[\/latex] for [latex]x[\/latex] near 1.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"342\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210811\/CNX_Calc_Figure_04_03_003.jpg\" alt=\"The function f(x) is shown, which curves upward from quadrant III, slows down in quadrant II, achieves a local maximum on the y-axis, decreases to achieve a local minimum in quadrant I at x = 1, increases to a local maximum at x = 2 that is greater than the other local maximum, and then decreases rapidly through quadrant IV.\" width=\"342\" height=\"395\" \/> Figure 3. This function [latex]f[\/latex] has two local maxima and one local minimum. The local maximum at [latex]x=2[\/latex] is also the absolute maximum.[\/caption]\r\n<div id=\"fs-id1165042275960\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165041836623\">A function [latex]f[\/latex] has a <strong>local maximum<\/strong> at [latex]c[\/latex] if there exists an open interval [latex]I[\/latex] containing [latex]c[\/latex] such that [latex]I[\/latex] is contained in the domain of [latex]f[\/latex] and [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex]. A function [latex]f[\/latex] has a <strong>local minimum<\/strong> at [latex]c[\/latex] if there exists an open interval [latex]I[\/latex] containing [latex]c[\/latex] such that [latex]I[\/latex] is contained in the domain of [latex]f[\/latex] and [latex]f(c)\\le f(x)[\/latex] for all [latex]x\\in I[\/latex]. A function [latex]f[\/latex] has a local extremum at [latex]c[\/latex] if [latex]f[\/latex] has a local maximum at [latex]c[\/latex] or [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165042007321\">Note that if [latex]f[\/latex] has an absolute extremum at [latex]c[\/latex] and [latex]f[\/latex] is defined over an interval containing [latex]c[\/latex], then [latex]f(c)[\/latex] is also considered a<strong> local extremum<\/strong>. If an absolute extremum for a function [latex]f[\/latex] occurs at an endpoint, we do not consider that to be a local extremum, but instead refer to that as an endpoint extremum.<\/p>\r\n<p id=\"fs-id1165041841707\">Given the graph of a function [latex]f[\/latex], it is sometimes easy to see where a local maximum or local minimum occurs. However, it is not always easy to see, since the interesting features on the graph of a function may not be visible because they occur at a very small scale. Also, we may not have a graph of the function. In these cases, how can we use a formula for a function to determine where these extrema occur?<\/p>\r\n<p id=\"fs-id1165042305291\">To answer this question, let\u2019s look at Figure 3 again. The local extrema occur at [latex]x=0[\/latex], [latex]x=1[\/latex], and [latex]x=2[\/latex]. Notice that at [latex]x=0[\/latex] and [latex]x=1[\/latex], the derivative [latex]f^{\\prime}(x)=0[\/latex]. At [latex]x=2[\/latex], the derivative [latex]f^{\\prime}(x)[\/latex] does not exist, since the function [latex]f[\/latex] has a corner there. In fact, if [latex]f[\/latex] has a local extremum at a point [latex]x=c[\/latex], the derivative [latex]f^{\\prime}(c)[\/latex] must satisfy one of the following conditions: either [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined. Such a value [latex]c[\/latex] is known as a<strong> critical point<\/strong> and it is important in finding extreme values for functions.<\/p>\r\n\r\n<div id=\"fs-id1165041782311\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165041779119\">Let [latex]c[\/latex] be an interior point in the domain of [latex]f[\/latex]. We say that [latex]c[\/latex] is a <strong>critical point<\/strong> of [latex]f[\/latex] if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165042038752\">As mentioned earlier, if [latex]f[\/latex] has a local extremum at a point [latex]x=c[\/latex], then [latex]c[\/latex] must be a critical point of [latex]f[\/latex]. This fact is known as<strong> Fermat\u2019s theorem.<\/strong><\/p>\r\n\r\n<div id=\"fs-id1165041796868\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Fermat\u2019s Theorem<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165041766544\">If [latex]f[\/latex] has a local extremum at [latex]c[\/latex] and [latex]f[\/latex] is differentiable at [latex]c[\/latex], then [latex]f^{\\prime}(c)=0[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165041842239\" class=\"bc-section section\">\r\n<h3>Proof<\/h3>\r\n<p id=\"fs-id1165041778659\">Suppose [latex]f[\/latex] has a local extremum at [latex]c[\/latex] and [latex]f[\/latex] is differentiable at [latex]c[\/latex]. We need to show that [latex]f^{\\prime}(c)=0[\/latex]. To do this, we will show that [latex]f^{\\prime}(c)\\ge 0[\/latex] and [latex]f^{\\prime}(c)\\le 0[\/latex], and therefore [latex]f^{\\prime}(c)=0[\/latex]. Since [latex]f[\/latex] has a local extremum at [latex]c[\/latex], [latex]f[\/latex] has a local maximum or local minimum at [latex]c[\/latex]. Suppose [latex]f[\/latex] has a local maximum at [latex]c[\/latex]. The case in which [latex]f[\/latex] has a local minimum at [latex]c[\/latex] can be handled similarly. There then exists an open interval [latex]I[\/latex] such that [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex]. Since [latex]f[\/latex] is differentiable at [latex]c[\/latex], from the definition of the derivative, we know that<\/p>\r\n\r\n<div id=\"fs-id1165042015585\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\underset{x\\to c}{\\lim}\\dfrac{f(x)-f(c)}{x-c}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165041888370\">Since this limit exists, both one-sided limits also exist and equal [latex]f^{\\prime}(c)[\/latex]. Therefore,<\/p>\r\n\r\n<div id=\"fs-id1165040744293\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\underset{x\\to c^+}{\\lim}\\dfrac{f(x)-f(c)}{x-c}[\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042015555\">and<\/p>\r\n\r\n<div id=\"fs-id1165042015558\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\underset{x\\to c^-}{\\lim}\\dfrac{f(x)-f(c)}{x-c}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1165042015667\">Since [latex]f(c)[\/latex] is a local maximum, we see that [latex]f(x)-f(c)\\le 0[\/latex] for [latex]x[\/latex] near [latex]c[\/latex]. Therefore, for [latex]x[\/latex] near [latex]c[\/latex], but [latex]x&gt;c[\/latex], we have [latex]\\frac{f(x)-f(c)}{x-c}\\le 0[\/latex]. From the equations above we conclude that [latex]f^{\\prime}(c)\\le 0[\/latex]. Similarly, it can be shown that [latex]f^{\\prime}(c)\\ge 0[\/latex]. Therefore, [latex]f^{\\prime}(c)=0[\/latex].<\/p>\r\n<p id=\"fs-id1165041832062\">[latex]_\\blacksquare[\/latex]<\/p>\r\n<p id=\"fs-id1165041832065\">From Fermat\u2019s theorem, we conclude that if [latex]f[\/latex] has a local extremum at [latex]c[\/latex], then either [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined. In other words, local extrema can only occur at critical points.<\/p>\r\n<p id=\"fs-id1165041832115\">Note this theorem does not claim that a function [latex]f[\/latex] must have a local extremum at a critical point. Rather, it states that critical points are candidates for local extrema. For example, consider the function [latex]f(x)=x^3[\/latex]. We have [latex]f^{\\prime}(x)=3x^2=0[\/latex] when [latex]x=0[\/latex]. Therefore, [latex]x=0[\/latex] is a critical point. However, [latex]f(x)=x^3[\/latex] is increasing over [latex](\u2212\\infty ,\\infty )[\/latex], and thus [latex]f[\/latex] does not have a local extremum at [latex]x=0[\/latex]. In Figure 4, we see several different possibilities for critical points. In some of these cases, the functions have local extrema at critical points, whereas in other cases the functions do not. Note that these graphs do not show all possibilities for the behavior of a function at a critical point.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210815\/CNX_Calc_Figure_04_03_004.jpg\" alt=\"This figure has five parts a, b, c, d, and e. In figure a, a parabola is shown facing down in quadrant I; there is a horizontal tangent line at the local maximum marked f\u2019(c) = 0. In figure b, there is a function drawn with an asymptote at c, meaning that the function increases toward infinity on both sides of c; it is noted that f\u2019(c) is undefined. In figure c, a version of the absolute value graph is shown that has been shifted so that its minimum is in quadrant I with x = c. It is noted that f\u2019(c) is undefined. In figure d, a version of the function f(x) = x3 is shown that has been shifted so that its inflection point is in quadrant I with x = c. Its inflection point at (c, f(c)) has a horizontal line through it, and it is noted that f\u2019(c) = 0. In figure e, a version of the function f(x) = x1\/3 is shown that has been shifted so that its inflection point is in quadrant I with x = c. Its inflection point at (c, f(c)) has a vertical line through it, and it is noted that f\u2019(c) is undefined.\" width=\"975\" height=\"563\" \/> Figure 4. (a\u2013e) A function [latex]f[\/latex] has a critical point at [latex]c[\/latex] if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined. A function may or may not have a local extremum at a critical point.[\/caption]\r\n<p id=\"fs-id1165041979108\">Later in this module we look at analytical methods for determining whether a function actually has a local extremum at a critical point. For now, let\u2019s turn our attention to finding critical points. We will use graphical observations to determine whether a critical point is associated with a local extremum.<\/p>\r\n\r\n<div id=\"fs-id1165041979119\" class=\"textbook exercises\">\r\n<h3>Example: Locating Critical Points<\/h3>\r\n<p id=\"fs-id1165041979129\">For each of the following functions, find all critical points. Use a graphing utility to determine whether the function has a local extremum at each of the critical points.<\/p>\r\n\r\n<ol id=\"fs-id1165041979134\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(x)=\\frac{1}{3}x^3-\\frac{5}{2}x^2+4x[\/latex]<\/li>\r\n \t<li>[latex]f(x)=(x^2-1)^3[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{4x}{1+x^2}[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165041979121\" class=\"exercise\">[reveal-answer q=\"fs-id1165041837084\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165041837084\"]\r\n<ol id=\"fs-id1165041837084\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>The derivative [latex]f^{\\prime}(x)=x^2-5x+4[\/latex] is defined for all real numbers [latex]x[\/latex]. Therefore, we only need to find the values for [latex]x[\/latex] where [latex]f^{\\prime}(x)=0[\/latex]. Since [latex]f^{\\prime}(x)=x^2-5x+4=(x-4)(x-1)[\/latex], the critical points are [latex]x=1[\/latex] and [latex]x=4[\/latex]. From the graph of [latex]f[\/latex] in Figure 5, we see that [latex]f[\/latex] has a local maximum at [latex]x=1[\/latex] and a local minimum at [latex]x=4[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210817\/CNX_Calc_Figure_04_03_005.jpg\" alt=\"The function f(x) = (1\/3) x3 \u2013 (5\/2) x2 + 4x is graphed. The function has local maximum at x = 1 and local minimum at x = 4.\" width=\"325\" height=\"235\" \/> Figure 5. This function has a local maximum and a local minimum.[\/caption]\r\n\r\n<div class=\"wp-caption-text\"><\/div><\/li>\r\n \t<li>Using the chain rule, we see the derivative is\r\n<div id=\"fs-id1165042226046\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=3(x^2-1)^2(2x)=6x(x^2-1)^2[\/latex]<\/div>\r\nTherefore, [latex]f[\/latex] has critical points when [latex]x=0[\/latex] and when [latex]x^2-1=0[\/latex]. We conclude that the critical points are [latex]x=0,\\pm 1[\/latex]. From the graph of [latex]f[\/latex] in Figure 6, we see that [latex]f[\/latex] has a local (and absolute) minimum at [latex]x=0[\/latex], but does not have a local extremum at [latex]x=1[\/latex] or [latex]x=-1[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210820\/CNX_Calc_Figure_04_03_006.jpg\" alt=\"The function f(x) = (x2 \u2212 1)3 is graphed. The function has local minimum at x = 0, and inflection points at x = \u00b11.\" width=\"487\" height=\"235\" \/> Figure 6. This function has three critical points: [latex]x=0[\/latex], [latex]x=1[\/latex], and [latex]x=-1[\/latex]. The function has a local (and absolute) minimum at [latex]x=0[\/latex], but does not have extrema at the other two critical points.[\/caption]\r\n<div class=\"wp-caption-text\"><\/div><\/li>\r\n \t<li>By the chain rule, we see that the derivative is\r\n<div id=\"fs-id1165042299459\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\dfrac{(1+x^2 \\cdot 4)-4x(2x)}{(1+x^2)^2}=\\dfrac{4-4x^2}{(1+x^2)^2}[\/latex]<\/div>\r\nThe derivative is defined everywhere. Therefore, we only need to find values for [latex]x[\/latex] where [latex]f^{\\prime}(x)=0[\/latex]. Solving [latex]f^{\\prime}(x)=0[\/latex], we see that [latex]4-4x^2=0[\/latex], which implies [latex]x=\\pm 1[\/latex]. Therefore, the critical points are [latex]x=\\pm 1[\/latex]. From the graph of [latex]f[\/latex] in Figure 7, we see that [latex]f[\/latex] has an absolute maximum at [latex]x=1[\/latex] and an absolute minimum at [latex]x=-1[\/latex]. Hence, [latex]f[\/latex] has a local maximum at [latex]x=1[\/latex] and a local minimum at [latex]x=-1[\/latex]. (Note that if [latex]f[\/latex] has an absolute extremum over an interval [latex]I[\/latex] at a point [latex]c[\/latex] that is not an endpoint of [latex]I[\/latex], then [latex]f[\/latex] has a local extremum at [latex]c[\/latex].)\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210822\/CNX_Calc_Figure_04_03_007.jpg\" alt=\"The function f(x) = 4x\/(1 + x2) is graphed. The function has local\/absolute maximum at x = 1 and local\/absolute minimum at x = \u22121.\" width=\"325\" height=\"272\" \/> Figure 7. This function has an absolute maximum and an absolute minimum.[\/caption]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Locating Critical Points.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/UBaioDQW1L8?controls=0&amp;start=629&amp;end=1122&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.3MaximaAndMinima629to1122_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.3 Maxima and Minima\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind all critical points for [latex]f(x)=x^3-\\frac{1}{2}x^2-2x+1[\/latex].\r\n\r\n[reveal-answer q=\"30077654\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"30077654\"]\r\n<p id=\"fs-id1165040757635\">Calculate [latex]f^{\\prime}(x)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165040757600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165040757600\"]\r\n\r\n[latex]x=-\\frac{2}{3}[\/latex], [latex]x=1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]207990[\/ohm_question]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165040757660\" class=\"bc-section section\">\r\n<h2>Locating Absolute Extrema<\/h2>\r\n<p id=\"fs-id1165040757666\">The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in Figure 2, one or both of these absolute extrema could occur at an endpoint. If an absolute extremum does not occur at an endpoint, however, it must occur at an <strong>interior point<\/strong>, in which case the absolute extremum is a local extremum. Therefore, by Fermat's Theorem, the point [latex]c[\/latex] at which the local extremum occurs must be a critical point. We summarize this result in the following theorem.<\/p>\r\n\r\n<div id=\"fs-id1165040757685\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Location of Absolute Extrema<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165040757692\">Let [latex]f[\/latex] be a continuous function over a closed, bounded interval [latex]I[\/latex]. The absolute maximum of [latex]f[\/latex] over [latex]I[\/latex] and the absolute minimum of [latex]f[\/latex] over [latex]I[\/latex] must occur at endpoints of [latex]I[\/latex] or at critical points of [latex]f[\/latex] in [latex]I[\/latex].<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165042035392\">With this idea in mind, let\u2019s examine a procedure for locating absolute extrema.<\/p>\r\n\r\n<div id=\"fs-id1165042035397\" class=\"textbox examples\">\r\n<h3>Problem-Solving Strategy: Locating Absolute Extrema over a Closed Interval<\/h3>\r\n<p id=\"fs-id1165042035403\">Consider a continuous function [latex]f[\/latex] defined over the closed interval [latex][a,b][\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1165042035429\">\r\n \t<li>Evaluate [latex]f[\/latex] at the endpoints [latex]x=a[\/latex] and [latex]x=b[\/latex].<\/li>\r\n \t<li>Find all critical points of [latex]f[\/latex] that lie over the interval [latex](a,b)[\/latex] and evaluate [latex]f[\/latex] at those critical points.<\/li>\r\n \t<li>Compare all values found in (1) and (2). From the location of absolute extrema, the absolute extrema must occur at endpoints or critical points. Therefore, the largest of these values is the absolute maximum of [latex]f[\/latex]. The smallest of these values is the absolute minimum of [latex]f[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1165042035511\">Now let\u2019s look at how to use this strategy to find the absolute maximum and absolute minimum values for continuous functions.<\/p>\r\n\r\n<div id=\"fs-id1165042035517\" class=\"textbook exercises\">\r\n<h3>Example: Locating Absolute Extrema<\/h3>\r\n<p id=\"fs-id1165042035527\">For each of the following functions, find the absolute maximum and absolute minimum over the specified interval and state where those values occur.<\/p>\r\n\r\n<ol id=\"fs-id1165042035532\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(x)=\u2212x^2+3x-2[\/latex] over [latex][1,3][\/latex].<\/li>\r\n \t<li>[latex]f(x)=x^2-3x^{\\frac{2}{3}}[\/latex] over [latex][0,2][\/latex].<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1165042035519\" class=\"exercise\">[reveal-answer q=\"fs-id1165042068485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042068485\"]\r\n<ol id=\"fs-id1165042068485\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>Step 1. Evaluate [latex]f[\/latex] at the endpoints [latex]x=1[\/latex] and [latex]x=3[\/latex].\r\n<div id=\"fs-id1165042068518\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(1)=0[\/latex] and [latex]f(3)=-2[\/latex]<\/div>\r\nStep 2. Since [latex]f^{\\prime}(x)=-2x+3[\/latex], [latex]f^{\\prime}[\/latex] is defined for all real numbers [latex]x[\/latex]. Therefore, there are no critical points where the derivative is undefined. It remains to check where [latex]f^{\\prime}(x)=0[\/latex]. Since [latex]f^{\\prime}(x)=-2x+3=0[\/latex] at [latex]x=\\frac{3}{2}[\/latex] and [latex]\\frac{3}{2}[\/latex] is in the interval [latex][1,3][\/latex], [latex]f(\\frac{3}{2})[\/latex] is a candidate for an absolute extremum of [latex]f[\/latex] over [latex][1,3][\/latex]. We evaluate [latex]f(\\frac{3}{2})[\/latex] and find\r\n<div id=\"fs-id1165042088374\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(\\frac{3}{2}\\right)=\\frac{1}{4}[\/latex]<\/div>\r\nStep 3. We set up the following table to compare the values found in steps 1 and 2.\r\n<table id=\"fs-id1165042088411\" class=\"unnumbered\" summary=\"This table has three columns and four rows. The first row is a header row, and it reads x, f(x), and Conclusion. Below the header row, the first column reads 0, 0, and blank. The second column reads 3\/2, 1\/4, and Absolute maximum. The third column reads 3, \u22122, and Absolute minimum.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{3}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\r\n<td>Absolute maximum<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>-2<\/td>\r\n<td>Absolute minimum<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom the table, we find that the absolute maximum of [latex]f[\/latex] over the interval [latex][1,3][\/latex] is [latex]\\frac{1}{4}[\/latex], and it occurs at [latex]x=\\frac{3}{2}[\/latex]. The absolute minimum of [latex]f[\/latex] over the interval [latex][1,3][\/latex] is -2, and it occurs at [latex]x=3[\/latex] as shown in the following graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210825\/CNX_Calc_Figure_04_03_008.jpg\" alt=\"The function f(x) = \u2013 x2 + 3x \u2013 2 is graphed from (1, 0) to (3, \u22122), with its maximum marked at (3\/2, 1\/4).\" width=\"731\" height=\"347\" \/> Figure 8. This function has both an absolute maximum and an absolute minimum.[\/caption]<\/li>\r\n \t<li>Step 1. Evaluate [latex]f[\/latex] at the endpoints [latex]x=0[\/latex] and [latex]x=2[\/latex].\r\n<div id=\"fs-id1165040744374\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(0)=0[\/latex] and [latex]f(2)=4-3\\sqrt[3]{4}\\approx -0.762[\/latex]<\/div>\r\nStep 2. The derivative of [latex]f[\/latex] is given by\r\n<div id=\"fs-id1165040744442\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=2x-\\dfrac{2}{x^{\\frac{1}{3}}}=\\dfrac{2x^{\\frac{4}{3}}-2}{x^{\\frac{1}{3}}}[\/latex]<\/div>\r\nfor [latex]x\\ne 0[\/latex]. The derivative is zero when [latex]2x^{\\frac{4}{3}}-2=0[\/latex], which implies [latex]x=\\pm 1[\/latex]. The derivative is undefined at [latex]x=0[\/latex]. Therefore, the critical points of [latex]f[\/latex] are [latex]x=0,1,-1[\/latex]. The point [latex]x=0[\/latex] is an endpoint, so we already evaluated [latex]f(0)[\/latex] in step 1. The point [latex]x=-1[\/latex] is not in the interval of interest, so we need only evaluate [latex]f(1)[\/latex]. We find that\r\n<div id=\"fs-id1165041848326\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(1)=-2[\/latex]<\/div>\r\nStep 3. We compare the values found in steps 1 and 2, in the following table.\r\n<table id=\"fs-id1165041848355\" class=\"unnumbered\" summary=\"This table has three columns and four rows. The first row is a header row, and it reads x, f(x), and Conclusion. Below the header row, the first column reads 0, 0, and Absolute maximum. The second column reads 1, \u22122, and Absolute minimum. The third column reads 2, \u22120.762, and blank.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f(x)[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<td>Absolute maximum<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>-2<\/td>\r\n<td>Absolute minimum<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>-0.762<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe conclude that the absolute maximum of [latex]f[\/latex] over the interval [latex][0,2][\/latex] is zero, and it occurs at [latex]x=0[\/latex]. The absolute minimum is \u22122, and it occurs at [latex]x=1[\/latex] as shown in the following graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"352\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210828\/CNX_Calc_Figure_04_03_009.jpg\" alt=\"The function f(x) = x2 \u2013 3x2\/3 is graphed from (0, 0) to (2, \u22120.762), with its minimum marked at (1, \u22122).\" width=\"352\" height=\"422\" \/> Figure 9. This function has an absolute maximum at an endpoint of the interval.[\/caption]\r\n\r\n<div class=\"wp-caption-text\"><\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Locating Absolute Extrema part (a).\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/UBaioDQW1L8?controls=0&amp;start=1123&amp;end=1272&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.3MaximaAndMinima1123to1272_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.3 Maxima and Minima\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the absolute maximum and absolute minimum of [latex]f(x)=x^2-4x+3[\/latex] over the interval [latex][1,4][\/latex].\r\n\r\n[reveal-answer q=\"4227188\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"4227188\"]\r\n\r\nLook for critical points. Evaluate [latex]f[\/latex] at all critical points and at the endpoints.\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165042108934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042108934\"]\r\n\r\nThe absolute maximum is 3 and it occurs at [latex]x=4[\/latex]. The absolute minimum is -1 and it occurs at [latex]x=2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165040729367\">At this point, we know how to locate absolute extrema for continuous functions over closed intervals. We have also defined local extrema and determined that if a function [latex]f[\/latex] has a local extremum at a point [latex]c[\/latex], then [latex]c[\/latex] must be a critical point of [latex]f[\/latex]. However, [latex]c[\/latex] being a critical point is not a sufficient condition for [latex]f[\/latex] to have a local extremum at [latex]c[\/latex]. Later in this module, we show how to determine whether a function actually has a local extremum at a critical point. First, however, we need to introduce the Mean Value Theorem, which will help as we analyze the behavior of the graph of a function.<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]5224[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define absolute extrema<\/li>\n<li>Define local extrema<\/li>\n<li>Explain how to find the critical points of a function over a closed interval<\/li>\n<li>Describe how to use critical points to locate absolute extrema over a closed interval<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165041932730\" class=\"bc-section section\">\n<h2>Absolute Extrema<\/h2>\n<p id=\"fs-id1165040644979\">Consider the function [latex]f(x)=x^2+1[\/latex] over the interval [latex](\u2212\\infty ,\\infty )[\/latex]. As [latex]x\\to \\pm \\infty[\/latex], [latex]f(x)\\to \\infty[\/latex]. Therefore, the function does not have a largest value. However, since [latex]x^2+1\\ge 1[\/latex] for all real numbers [latex]x[\/latex] and [latex]x^2+1=1[\/latex] when [latex]x=0[\/latex], the function has a smallest value, 1, when [latex]x=0[\/latex]. We say that 1 is the<strong> absolute minimum<\/strong> of [latex]f(x)=x^2+1[\/latex] and it occurs at [latex]x=0[\/latex]. We say that [latex]f(x)=x^2+1[\/latex] does not have an<strong> absolute maximum<\/strong> (see Figure 1).<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210800\/CNX_Calc_Figure_04_03_001.jpg\" alt=\"The function f(x) = x2 + 1 is graphed, and its minimum of 1 is seen to be at x = 0.\" width=\"487\" height=\"271\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. The given function has an absolute minimum of 1 at [latex]x=0[\/latex]. The function does not have an absolute maximum.<\/p>\n<\/div>\n<div id=\"fs-id1165041992939\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1165041930563\">Let [latex]f[\/latex] be a function defined over an interval [latex]I[\/latex] and let [latex]c\\in I[\/latex]. We say [latex]f[\/latex] has an <strong>absolute maximum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex] if [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex]. We say [latex]f[\/latex] has an <strong>absolute minimum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex] if [latex]f(c)\\le f(x)[\/latex] for all [latex]x\\in I[\/latex]. If [latex]f[\/latex] has an absolute maximum on [latex]I[\/latex] at [latex]c[\/latex] or an absolute minimum on [latex]I[\/latex] at [latex]c[\/latex], we say [latex]f[\/latex] has an<strong> absolute extremum<\/strong> on [latex]I[\/latex] at [latex]c[\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1165041811976\">Before proceeding, let\u2019s note two important issues regarding this definition. First, the term <em>absolute<\/em> here does not refer to absolute value. An absolute extremum may be positive, negative, or zero. Second, if a function [latex]f[\/latex] has an absolute extremum over an interval [latex]I[\/latex] at [latex]c[\/latex], the absolute extremum is [latex]f(c)[\/latex]. The real number [latex]c[\/latex] is a point in the domain at which the absolute extremum occurs. For example, consider the function [latex]f(x)=\\frac{1}{(x^2+1)}[\/latex] over the interval [latex](\u2212\\infty ,\\infty )[\/latex]. Since<\/p>\n<div id=\"fs-id1165041761810\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(0)=1\\ge \\dfrac{1}{x^2+1}=f(x)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165041952862\">for all real numbers [latex]x[\/latex], we say [latex]f[\/latex] has an absolute maximum over [latex](\u2212\\infty ,\\infty )[\/latex] at [latex]x=0[\/latex]. The absolute maximum is [latex]f(0)=1[\/latex]. It occurs at [latex]x=0[\/latex], as shown in Figure 2b.<\/p>\n<p>So remember: the maximum\/minimum = [latex]y[\/latex]; the location of the maximum\/minimum =\u00a0[latex]x[\/latex].<\/p>\n<p id=\"fs-id1165041795038\">A function may have both an absolute maximum and an absolute minimum, just one extremum, or neither. Figure 2 shows several functions and some of the different possibilities regarding absolute extrema. However, the following theorem, called the<strong> Extreme Value Theorem<\/strong>, guarantees that a continuous function [latex]f[\/latex] over a closed, bounded interval [latex][a,b][\/latex] has both an absolute maximum and an absolute minimum.<\/p>\n<div style=\"width: 933px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210805\/CNX_Calc_Figure_04_03_010.jpg\" alt=\"This figure has six parts a, b, c, d, e, and f. In figure a, the line f(x) = x3 is shown, and it is noted that it has no absolute minimum and no absolute maximum. In figure b, the line f(x) = 1\/(x2 + 1) is shown, which is near 0 for most of its length and rises to a bump at (0, 1); it has no absolute minimum, but does have an absolute maximum of 1 at x = 0. In figure c, the line f(x) = cos x is shown, which has absolute minimums of \u22121 at \u00b1\u03c0, \u00b13\u03c0, \u2026 and absolute maximums of 1 at 0, \u00b12\u03c0, \u00b14\u03c0, \u2026. In figure d, the piecewise function f(x) = 2 \u2013 x2 for 0 \u2264 x &lt; 2 and x \u2013 3 for 2 \u2264 x \u2264 4 is shown, with absolute maximum of 2 at x = 0 and no absolute minimum. In figure e, the function f(x) = (x \u2013 2)2 is shown on [1, 4], which has absolute maximum of 4 at x = 4 and absolute minimum of 0 at x = 2. In figure f, the function f(x) = x\/(2 \u2212 x) is shown on [0, 2), with absolute minimum of 0 at x = 0 and no absolute maximum.\" width=\"923\" height=\"900\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. Graphs (a), (b), and (c) show several possibilities for absolute extrema for functions with a domain of [latex](\u2212\\infty ,\\infty )[\/latex]. Graphs (d), (e), and (f) show several possibilities for absolute extrema for functions with a domain that is a bounded interval.<\/p>\n<\/div>\n<div id=\"fs-id1165042118849\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Extreme Value Theorem<\/h3>\n<hr \/>\n<p id=\"fs-id1165042274290\">If [latex]f[\/latex] is a continuous function over the closed, bounded interval [latex][a,b][\/latex], then there is a point in [latex][a,b][\/latex] at which [latex]f[\/latex] has an absolute maximum over [latex][a,b][\/latex] and there is a point in [latex][a,b][\/latex] at which [latex]f[\/latex] has an absolute minimum over [latex][a,b][\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1165040759275\">The proof of the extreme value theorem is beyond the scope of this text. Typically, it is proved in a course on real analysis. There are a couple of key points to note about the statement of this theorem. For the extreme value theorem to apply, the function must be continuous over a closed, bounded interval. If the interval [latex]I[\/latex] is open or the function has even one point of discontinuity, the function may not have an absolute maximum or absolute minimum over [latex]I[\/latex]. For example, consider the functions shown in Figure 2(d), (e), and (f). All three of these functions are defined over bounded intervals. However, the function in graph (e) is the only one that has both an absolute maximum and an absolute minimum over its domain. The extreme value theorem cannot be applied to the functions in graphs (d) and (f) because neither of these functions is continuous over a closed, bounded interval. Although the function in graph (d) is defined over the closed interval [latex][0,4][\/latex], the function is discontinuous at [latex]x=2[\/latex]. The function has an absolute maximum over [latex][0,4][\/latex] but does not have an absolute minimum. The function in graph (f) is continuous over the half-open interval [latex][0,2)[\/latex], but is not defined at [latex]x=2[\/latex], and therefore is not continuous over a closed, bounded interval. The function has an absolute minimum over [latex][0,2)[\/latex], but does not have an absolute maximum over [latex][0,2)[\/latex]. These two graphs illustrate why a function over a bounded interval may fail to have an absolute maximum and\/or absolute minimum.<\/p>\n<p id=\"fs-id1165041805532\">Before looking at how to find absolute extrema, let\u2019s examine the related concept of local extrema. This idea is useful in determining where absolute extrema occur.<\/p>\n<\/div>\n<h2>Local Extrema and Critical Points<\/h2>\n<p id=\"fs-id1165042120195\">Consider the function [latex]f[\/latex] shown in Figure 3. The graph can be described as two mountains with a valley in the middle. The absolute maximum value of the function occurs at the higher peak, at [latex]x=2[\/latex]. However, [latex]x=0[\/latex] is also a point of interest. Although [latex]f(0)[\/latex] is not the largest value of [latex]f[\/latex], the value [latex]f(0)[\/latex] is larger than [latex]f(x)[\/latex] for all [latex]x[\/latex] near 0. We say [latex]f[\/latex] has a local maximum at [latex]x=0[\/latex]. Similarly, the function [latex]f[\/latex] does not have an absolute minimum, but it does have a local minimum at [latex]x=1[\/latex] because [latex]f(1)[\/latex] is less than [latex]f(x)[\/latex] for [latex]x[\/latex] near 1.<\/p>\n<div style=\"width: 352px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210811\/CNX_Calc_Figure_04_03_003.jpg\" alt=\"The function f(x) is shown, which curves upward from quadrant III, slows down in quadrant II, achieves a local maximum on the y-axis, decreases to achieve a local minimum in quadrant I at x = 1, increases to a local maximum at x = 2 that is greater than the other local maximum, and then decreases rapidly through quadrant IV.\" width=\"342\" height=\"395\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. This function [latex]f[\/latex] has two local maxima and one local minimum. The local maximum at [latex]x=2[\/latex] is also the absolute maximum.<\/p>\n<\/div>\n<div id=\"fs-id1165042275960\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1165041836623\">A function [latex]f[\/latex] has a <strong>local maximum<\/strong> at [latex]c[\/latex] if there exists an open interval [latex]I[\/latex] containing [latex]c[\/latex] such that [latex]I[\/latex] is contained in the domain of [latex]f[\/latex] and [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex]. A function [latex]f[\/latex] has a <strong>local minimum<\/strong> at [latex]c[\/latex] if there exists an open interval [latex]I[\/latex] containing [latex]c[\/latex] such that [latex]I[\/latex] is contained in the domain of [latex]f[\/latex] and [latex]f(c)\\le f(x)[\/latex] for all [latex]x\\in I[\/latex]. A function [latex]f[\/latex] has a local extremum at [latex]c[\/latex] if [latex]f[\/latex] has a local maximum at [latex]c[\/latex] or [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1165042007321\">Note that if [latex]f[\/latex] has an absolute extremum at [latex]c[\/latex] and [latex]f[\/latex] is defined over an interval containing [latex]c[\/latex], then [latex]f(c)[\/latex] is also considered a<strong> local extremum<\/strong>. If an absolute extremum for a function [latex]f[\/latex] occurs at an endpoint, we do not consider that to be a local extremum, but instead refer to that as an endpoint extremum.<\/p>\n<p id=\"fs-id1165041841707\">Given the graph of a function [latex]f[\/latex], it is sometimes easy to see where a local maximum or local minimum occurs. However, it is not always easy to see, since the interesting features on the graph of a function may not be visible because they occur at a very small scale. Also, we may not have a graph of the function. In these cases, how can we use a formula for a function to determine where these extrema occur?<\/p>\n<p id=\"fs-id1165042305291\">To answer this question, let\u2019s look at Figure 3 again. The local extrema occur at [latex]x=0[\/latex], [latex]x=1[\/latex], and [latex]x=2[\/latex]. Notice that at [latex]x=0[\/latex] and [latex]x=1[\/latex], the derivative [latex]f^{\\prime}(x)=0[\/latex]. At [latex]x=2[\/latex], the derivative [latex]f^{\\prime}(x)[\/latex] does not exist, since the function [latex]f[\/latex] has a corner there. In fact, if [latex]f[\/latex] has a local extremum at a point [latex]x=c[\/latex], the derivative [latex]f^{\\prime}(c)[\/latex] must satisfy one of the following conditions: either [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined. Such a value [latex]c[\/latex] is known as a<strong> critical point<\/strong> and it is important in finding extreme values for functions.<\/p>\n<div id=\"fs-id1165041782311\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1165041779119\">Let [latex]c[\/latex] be an interior point in the domain of [latex]f[\/latex]. We say that [latex]c[\/latex] is a <strong>critical point<\/strong> of [latex]f[\/latex] if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined.<\/p>\n<\/div>\n<p id=\"fs-id1165042038752\">As mentioned earlier, if [latex]f[\/latex] has a local extremum at a point [latex]x=c[\/latex], then [latex]c[\/latex] must be a critical point of [latex]f[\/latex]. This fact is known as<strong> Fermat\u2019s theorem.<\/strong><\/p>\n<div id=\"fs-id1165041796868\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Fermat\u2019s Theorem<\/h3>\n<hr \/>\n<p id=\"fs-id1165041766544\">If [latex]f[\/latex] has a local extremum at [latex]c[\/latex] and [latex]f[\/latex] is differentiable at [latex]c[\/latex], then [latex]f^{\\prime}(c)=0[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165041842239\" class=\"bc-section section\">\n<h3>Proof<\/h3>\n<p id=\"fs-id1165041778659\">Suppose [latex]f[\/latex] has a local extremum at [latex]c[\/latex] and [latex]f[\/latex] is differentiable at [latex]c[\/latex]. We need to show that [latex]f^{\\prime}(c)=0[\/latex]. To do this, we will show that [latex]f^{\\prime}(c)\\ge 0[\/latex] and [latex]f^{\\prime}(c)\\le 0[\/latex], and therefore [latex]f^{\\prime}(c)=0[\/latex]. Since [latex]f[\/latex] has a local extremum at [latex]c[\/latex], [latex]f[\/latex] has a local maximum or local minimum at [latex]c[\/latex]. Suppose [latex]f[\/latex] has a local maximum at [latex]c[\/latex]. The case in which [latex]f[\/latex] has a local minimum at [latex]c[\/latex] can be handled similarly. There then exists an open interval [latex]I[\/latex] such that [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex]. Since [latex]f[\/latex] is differentiable at [latex]c[\/latex], from the definition of the derivative, we know that<\/p>\n<div id=\"fs-id1165042015585\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\underset{x\\to c}{\\lim}\\dfrac{f(x)-f(c)}{x-c}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165041888370\">Since this limit exists, both one-sided limits also exist and equal [latex]f^{\\prime}(c)[\/latex]. Therefore,<\/p>\n<div id=\"fs-id1165040744293\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\underset{x\\to c^+}{\\lim}\\dfrac{f(x)-f(c)}{x-c}[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042015555\">and<\/p>\n<div id=\"fs-id1165042015558\" class=\"equation\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\underset{x\\to c^-}{\\lim}\\dfrac{f(x)-f(c)}{x-c}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1165042015667\">Since [latex]f(c)[\/latex] is a local maximum, we see that [latex]f(x)-f(c)\\le 0[\/latex] for [latex]x[\/latex] near [latex]c[\/latex]. Therefore, for [latex]x[\/latex] near [latex]c[\/latex], but [latex]x>c[\/latex], we have [latex]\\frac{f(x)-f(c)}{x-c}\\le 0[\/latex]. From the equations above we conclude that [latex]f^{\\prime}(c)\\le 0[\/latex]. Similarly, it can be shown that [latex]f^{\\prime}(c)\\ge 0[\/latex]. Therefore, [latex]f^{\\prime}(c)=0[\/latex].<\/p>\n<p id=\"fs-id1165041832062\">[latex]_\\blacksquare[\/latex]<\/p>\n<p id=\"fs-id1165041832065\">From Fermat\u2019s theorem, we conclude that if [latex]f[\/latex] has a local extremum at [latex]c[\/latex], then either [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined. In other words, local extrema can only occur at critical points.<\/p>\n<p id=\"fs-id1165041832115\">Note this theorem does not claim that a function [latex]f[\/latex] must have a local extremum at a critical point. Rather, it states that critical points are candidates for local extrema. For example, consider the function [latex]f(x)=x^3[\/latex]. We have [latex]f^{\\prime}(x)=3x^2=0[\/latex] when [latex]x=0[\/latex]. Therefore, [latex]x=0[\/latex] is a critical point. However, [latex]f(x)=x^3[\/latex] is increasing over [latex](\u2212\\infty ,\\infty )[\/latex], and thus [latex]f[\/latex] does not have a local extremum at [latex]x=0[\/latex]. In Figure 4, we see several different possibilities for critical points. In some of these cases, the functions have local extrema at critical points, whereas in other cases the functions do not. Note that these graphs do not show all possibilities for the behavior of a function at a critical point.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210815\/CNX_Calc_Figure_04_03_004.jpg\" alt=\"This figure has five parts a, b, c, d, and e. In figure a, a parabola is shown facing down in quadrant I; there is a horizontal tangent line at the local maximum marked f\u2019(c) = 0. In figure b, there is a function drawn with an asymptote at c, meaning that the function increases toward infinity on both sides of c; it is noted that f\u2019(c) is undefined. In figure c, a version of the absolute value graph is shown that has been shifted so that its minimum is in quadrant I with x = c. It is noted that f\u2019(c) is undefined. In figure d, a version of the function f(x) = x3 is shown that has been shifted so that its inflection point is in quadrant I with x = c. Its inflection point at (c, f(c)) has a horizontal line through it, and it is noted that f\u2019(c) = 0. In figure e, a version of the function f(x) = x1\/3 is shown that has been shifted so that its inflection point is in quadrant I with x = c. Its inflection point at (c, f(c)) has a vertical line through it, and it is noted that f\u2019(c) is undefined.\" width=\"975\" height=\"563\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. (a\u2013e) A function [latex]f[\/latex] has a critical point at [latex]c[\/latex] if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined. A function may or may not have a local extremum at a critical point.<\/p>\n<\/div>\n<p id=\"fs-id1165041979108\">Later in this module we look at analytical methods for determining whether a function actually has a local extremum at a critical point. For now, let\u2019s turn our attention to finding critical points. We will use graphical observations to determine whether a critical point is associated with a local extremum.<\/p>\n<div id=\"fs-id1165041979119\" class=\"textbook exercises\">\n<h3>Example: Locating Critical Points<\/h3>\n<p id=\"fs-id1165041979129\">For each of the following functions, find all critical points. Use a graphing utility to determine whether the function has a local extremum at each of the critical points.<\/p>\n<ol id=\"fs-id1165041979134\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(x)=\\frac{1}{3}x^3-\\frac{5}{2}x^2+4x[\/latex]<\/li>\n<li>[latex]f(x)=(x^2-1)^3[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{4x}{1+x^2}[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1165041979121\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165041837084\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165041837084\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165041837084\" style=\"list-style-type: lower-alpha;\">\n<li>The derivative [latex]f^{\\prime}(x)=x^2-5x+4[\/latex] is defined for all real numbers [latex]x[\/latex]. Therefore, we only need to find the values for [latex]x[\/latex] where [latex]f^{\\prime}(x)=0[\/latex]. Since [latex]f^{\\prime}(x)=x^2-5x+4=(x-4)(x-1)[\/latex], the critical points are [latex]x=1[\/latex] and [latex]x=4[\/latex]. From the graph of [latex]f[\/latex] in Figure 5, we see that [latex]f[\/latex] has a local maximum at [latex]x=1[\/latex] and a local minimum at [latex]x=4[\/latex].\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210817\/CNX_Calc_Figure_04_03_005.jpg\" alt=\"The function f(x) = (1\/3) x3 \u2013 (5\/2) x2 + 4x is graphed. The function has local maximum at x = 1 and local minimum at x = 4.\" width=\"325\" height=\"235\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. This function has a local maximum and a local minimum.<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<\/li>\n<li>Using the chain rule, we see the derivative is\n<div id=\"fs-id1165042226046\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=3(x^2-1)^2(2x)=6x(x^2-1)^2[\/latex]<\/div>\n<p>Therefore, [latex]f[\/latex] has critical points when [latex]x=0[\/latex] and when [latex]x^2-1=0[\/latex]. We conclude that the critical points are [latex]x=0,\\pm 1[\/latex]. From the graph of [latex]f[\/latex] in Figure 6, we see that [latex]f[\/latex] has a local (and absolute) minimum at [latex]x=0[\/latex], but does not have a local extremum at [latex]x=1[\/latex] or [latex]x=-1[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210820\/CNX_Calc_Figure_04_03_006.jpg\" alt=\"The function f(x) = (x2 \u2212 1)3 is graphed. The function has local minimum at x = 0, and inflection points at x = \u00b11.\" width=\"487\" height=\"235\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6. This function has three critical points: [latex]x=0[\/latex], [latex]x=1[\/latex], and [latex]x=-1[\/latex]. The function has a local (and absolute) minimum at [latex]x=0[\/latex], but does not have extrema at the other two critical points.<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<\/li>\n<li>By the chain rule, we see that the derivative is\n<div id=\"fs-id1165042299459\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\dfrac{(1+x^2 \\cdot 4)-4x(2x)}{(1+x^2)^2}=\\dfrac{4-4x^2}{(1+x^2)^2}[\/latex]<\/div>\n<p>The derivative is defined everywhere. Therefore, we only need to find values for [latex]x[\/latex] where [latex]f^{\\prime}(x)=0[\/latex]. Solving [latex]f^{\\prime}(x)=0[\/latex], we see that [latex]4-4x^2=0[\/latex], which implies [latex]x=\\pm 1[\/latex]. Therefore, the critical points are [latex]x=\\pm 1[\/latex]. From the graph of [latex]f[\/latex] in Figure 7, we see that [latex]f[\/latex] has an absolute maximum at [latex]x=1[\/latex] and an absolute minimum at [latex]x=-1[\/latex]. Hence, [latex]f[\/latex] has a local maximum at [latex]x=1[\/latex] and a local minimum at [latex]x=-1[\/latex]. (Note that if [latex]f[\/latex] has an absolute extremum over an interval [latex]I[\/latex] at a point [latex]c[\/latex] that is not an endpoint of [latex]I[\/latex], then [latex]f[\/latex] has a local extremum at [latex]c[\/latex].)<\/p>\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210822\/CNX_Calc_Figure_04_03_007.jpg\" alt=\"The function f(x) = 4x\/(1 + x2) is graphed. The function has local\/absolute maximum at x = 1 and local\/absolute minimum at x = \u22121.\" width=\"325\" height=\"272\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7. This function has an absolute maximum and an absolute minimum.<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Locating Critical Points.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/UBaioDQW1L8?controls=0&amp;start=629&amp;end=1122&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.3MaximaAndMinima629to1122_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.3 Maxima and Minima&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find all critical points for [latex]f(x)=x^3-\\frac{1}{2}x^2-2x+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q30077654\">Hint<\/span><\/p>\n<div id=\"q30077654\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165040757635\">Calculate [latex]f^{\\prime}(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165040757600\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165040757600\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-\\frac{2}{3}[\/latex], [latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm207990\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=207990&theme=oea&iframe_resize_id=ohm207990&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div id=\"fs-id1165040757660\" class=\"bc-section section\">\n<h2>Locating Absolute Extrema<\/h2>\n<p id=\"fs-id1165040757666\">The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in Figure 2, one or both of these absolute extrema could occur at an endpoint. If an absolute extremum does not occur at an endpoint, however, it must occur at an <strong>interior point<\/strong>, in which case the absolute extremum is a local extremum. Therefore, by Fermat&#8217;s Theorem, the point [latex]c[\/latex] at which the local extremum occurs must be a critical point. We summarize this result in the following theorem.<\/p>\n<div id=\"fs-id1165040757685\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Location of Absolute Extrema<\/h3>\n<hr \/>\n<p id=\"fs-id1165040757692\">Let [latex]f[\/latex] be a continuous function over a closed, bounded interval [latex]I[\/latex]. The absolute maximum of [latex]f[\/latex] over [latex]I[\/latex] and the absolute minimum of [latex]f[\/latex] over [latex]I[\/latex] must occur at endpoints of [latex]I[\/latex] or at critical points of [latex]f[\/latex] in [latex]I[\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1165042035392\">With this idea in mind, let\u2019s examine a procedure for locating absolute extrema.<\/p>\n<div id=\"fs-id1165042035397\" class=\"textbox examples\">\n<h3>Problem-Solving Strategy: Locating Absolute Extrema over a Closed Interval<\/h3>\n<p id=\"fs-id1165042035403\">Consider a continuous function [latex]f[\/latex] defined over the closed interval [latex][a,b][\/latex].<\/p>\n<ol id=\"fs-id1165042035429\">\n<li>Evaluate [latex]f[\/latex] at the endpoints [latex]x=a[\/latex] and [latex]x=b[\/latex].<\/li>\n<li>Find all critical points of [latex]f[\/latex] that lie over the interval [latex](a,b)[\/latex] and evaluate [latex]f[\/latex] at those critical points.<\/li>\n<li>Compare all values found in (1) and (2). From the location of absolute extrema, the absolute extrema must occur at endpoints or critical points. Therefore, the largest of these values is the absolute maximum of [latex]f[\/latex]. The smallest of these values is the absolute minimum of [latex]f[\/latex].<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1165042035511\">Now let\u2019s look at how to use this strategy to find the absolute maximum and absolute minimum values for continuous functions.<\/p>\n<div id=\"fs-id1165042035517\" class=\"textbook exercises\">\n<h3>Example: Locating Absolute Extrema<\/h3>\n<p id=\"fs-id1165042035527\">For each of the following functions, find the absolute maximum and absolute minimum over the specified interval and state where those values occur.<\/p>\n<ol id=\"fs-id1165042035532\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(x)=\u2212x^2+3x-2[\/latex] over [latex][1,3][\/latex].<\/li>\n<li>[latex]f(x)=x^2-3x^{\\frac{2}{3}}[\/latex] over [latex][0,2][\/latex].<\/li>\n<\/ol>\n<div id=\"fs-id1165042035519\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042068485\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042068485\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165042068485\" style=\"list-style-type: lower-alpha;\">\n<li>Step 1. Evaluate [latex]f[\/latex] at the endpoints [latex]x=1[\/latex] and [latex]x=3[\/latex].\n<div id=\"fs-id1165042068518\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(1)=0[\/latex] and [latex]f(3)=-2[\/latex]<\/div>\n<p>Step 2. Since [latex]f^{\\prime}(x)=-2x+3[\/latex], [latex]f^{\\prime}[\/latex] is defined for all real numbers [latex]x[\/latex]. Therefore, there are no critical points where the derivative is undefined. It remains to check where [latex]f^{\\prime}(x)=0[\/latex]. Since [latex]f^{\\prime}(x)=-2x+3=0[\/latex] at [latex]x=\\frac{3}{2}[\/latex] and [latex]\\frac{3}{2}[\/latex] is in the interval [latex][1,3][\/latex], [latex]f(\\frac{3}{2})[\/latex] is a candidate for an absolute extremum of [latex]f[\/latex] over [latex][1,3][\/latex]. We evaluate [latex]f(\\frac{3}{2})[\/latex] and find<\/p>\n<div id=\"fs-id1165042088374\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f\\left(\\frac{3}{2}\\right)=\\frac{1}{4}[\/latex]<\/div>\n<p>Step 3. We set up the following table to compare the values found in steps 1 and 2.<\/p>\n<table id=\"fs-id1165042088411\" class=\"unnumbered\" summary=\"This table has three columns and four rows. The first row is a header row, and it reads x, f(x), and Conclusion. Below the header row, the first column reads 0, 0, and blank. The second column reads 3\/2, 1\/4, and Absolute maximum. The third column reads 3, \u22122, and Absolute minimum.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{3}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>Absolute maximum<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>-2<\/td>\n<td>Absolute minimum<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From the table, we find that the absolute maximum of [latex]f[\/latex] over the interval [latex][1,3][\/latex] is [latex]\\frac{1}{4}[\/latex], and it occurs at [latex]x=\\frac{3}{2}[\/latex]. The absolute minimum of [latex]f[\/latex] over the interval [latex][1,3][\/latex] is -2, and it occurs at [latex]x=3[\/latex] as shown in the following graph.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210825\/CNX_Calc_Figure_04_03_008.jpg\" alt=\"The function f(x) = \u2013 x2 + 3x \u2013 2 is graphed from (1, 0) to (3, \u22122), with its maximum marked at (3\/2, 1\/4).\" width=\"731\" height=\"347\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8. This function has both an absolute maximum and an absolute minimum.<\/p>\n<\/div>\n<\/li>\n<li>Step 1. Evaluate [latex]f[\/latex] at the endpoints [latex]x=0[\/latex] and [latex]x=2[\/latex].\n<div id=\"fs-id1165040744374\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(0)=0[\/latex] and [latex]f(2)=4-3\\sqrt[3]{4}\\approx -0.762[\/latex]<\/div>\n<p>Step 2. The derivative of [latex]f[\/latex] is given by<\/p>\n<div id=\"fs-id1165040744442\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=2x-\\dfrac{2}{x^{\\frac{1}{3}}}=\\dfrac{2x^{\\frac{4}{3}}-2}{x^{\\frac{1}{3}}}[\/latex]<\/div>\n<p>for [latex]x\\ne 0[\/latex]. The derivative is zero when [latex]2x^{\\frac{4}{3}}-2=0[\/latex], which implies [latex]x=\\pm 1[\/latex]. The derivative is undefined at [latex]x=0[\/latex]. Therefore, the critical points of [latex]f[\/latex] are [latex]x=0,1,-1[\/latex]. The point [latex]x=0[\/latex] is an endpoint, so we already evaluated [latex]f(0)[\/latex] in step 1. The point [latex]x=-1[\/latex] is not in the interval of interest, so we need only evaluate [latex]f(1)[\/latex]. We find that<\/p>\n<div id=\"fs-id1165041848326\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(1)=-2[\/latex]<\/div>\n<p>Step 3. We compare the values found in steps 1 and 2, in the following table.<\/p>\n<table id=\"fs-id1165041848355\" class=\"unnumbered\" summary=\"This table has three columns and four rows. The first row is a header row, and it reads x, f(x), and Conclusion. Below the header row, the first column reads 0, 0, and Absolute maximum. The second column reads 1, \u22122, and Absolute minimum. The third column reads 2, \u22120.762, and blank.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f(x)[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<td>Absolute maximum<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>-2<\/td>\n<td>Absolute minimum<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>-0.762<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We conclude that the absolute maximum of [latex]f[\/latex] over the interval [latex][0,2][\/latex] is zero, and it occurs at [latex]x=0[\/latex]. The absolute minimum is \u22122, and it occurs at [latex]x=1[\/latex] as shown in the following graph.<\/p>\n<div style=\"width: 362px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210828\/CNX_Calc_Figure_04_03_009.jpg\" alt=\"The function f(x) = x2 \u2013 3x2\/3 is graphed from (0, 0) to (2, \u22120.762), with its minimum marked at (1, \u22122).\" width=\"352\" height=\"422\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9. This function has an absolute maximum at an endpoint of the interval.<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Locating Absolute Extrema part (a).<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/UBaioDQW1L8?controls=0&amp;start=1123&amp;end=1272&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.3MaximaAndMinima1123to1272_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.3 Maxima and Minima&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the absolute maximum and absolute minimum of [latex]f(x)=x^2-4x+3[\/latex] over the interval [latex][1,4][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4227188\">Hint<\/span><\/p>\n<div id=\"q4227188\" class=\"hidden-answer\" style=\"display: none\">\n<p>Look for critical points. Evaluate [latex]f[\/latex] at all critical points and at the endpoints.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042108934\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042108934\" class=\"hidden-answer\" style=\"display: none\">\n<p>The absolute maximum is 3 and it occurs at [latex]x=4[\/latex]. The absolute minimum is -1 and it occurs at [latex]x=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165040729367\">At this point, we know how to locate absolute extrema for continuous functions over closed intervals. We have also defined local extrema and determined that if a function [latex]f[\/latex] has a local extremum at a point [latex]c[\/latex], then [latex]c[\/latex] must be a critical point of [latex]f[\/latex]. However, [latex]c[\/latex] being a critical point is not a sufficient condition for [latex]f[\/latex] to have a local extremum at [latex]c[\/latex]. Later in this module, we show how to determine whether a function actually has a local extremum at a critical point. First, however, we need to introduce the Mean Value Theorem, which will help as we analyze the behavior of the graph of a function.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm5224\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=5224&theme=oea&iframe_resize_id=ohm5224&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-399\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>4.3 Maxima and Minima. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"4.3 Maxima and Minima\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-399","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/399","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":26,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/399\/revisions"}],"predecessor-version":[{"id":4831,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/399\/revisions\/4831"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/399\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=399"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=399"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=399"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}