{"id":404,"date":"2021-02-04T02:01:08","date_gmt":"2021-02-04T02:01:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=404"},"modified":"2022-03-16T05:46:15","modified_gmt":"2022-03-16T05:46:15","slug":"the-second-derivative-test","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-second-derivative-test\/","title":{"raw":"The Second Derivative Test","rendered":"The Second Derivative Test"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Explain the relationship between a function and its first and second derivatives<\/li>\r\n \t<li>State the second derivative test for local extrema<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165043423622\" class=\"bc-section section\">\r\n<p id=\"fs-id1165042621320\">The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. Using the second derivative can sometimes be a simpler method than using the first derivative.<\/p>\r\n<p id=\"fs-id1165042621325\">We know that if a continuous function has a local extrema, it must occur at a critical point. However, a function need not have a local extrema at a critical point. Here we examine how the<strong> second derivative test<\/strong> can be used to determine whether a function has a local extremum at a critical point. Let [latex]f[\/latex] be a twice-differentiable function such that [latex]f^{\\prime}(a)=0[\/latex] and [latex]f^{\\prime \\prime}[\/latex] is continuous over an open interval [latex]I[\/latex] containing [latex]a[\/latex]. Suppose [latex]f^{\\prime \\prime}(a)&lt;0[\/latex]. Since [latex]f^{\\prime \\prime}[\/latex] is continuous over [latex]I[\/latex], [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] for all [latex]x \\in I[\/latex] (Figure 9). Then, by Corollary 3, [latex]f^{\\prime}[\/latex] is a decreasing function over [latex]I[\/latex]. Since [latex]f^{\\prime}(a)=0[\/latex], we conclude that for all [latex]x \\in I, \\, f^{\\prime}(x)&gt;0[\/latex] if [latex]x&lt;a[\/latex] and [latex]f^{\\prime}(x)&lt;0[\/latex] if [latex]x&gt;a[\/latex]. Therefore, by the first derivative test, [latex]f[\/latex] has a local maximum at [latex]x=a[\/latex]. On the other hand, suppose there exists a point [latex]b[\/latex] such that [latex]f^{\\prime}(b)=0[\/latex] but [latex]f^{\\prime \\prime}(b)&gt;0[\/latex]. Since [latex]f^{\\prime \\prime}[\/latex] is continuous over an open interval [latex]I[\/latex] containing [latex]b[\/latex], then [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] for all [latex]x \\in I[\/latex] (Figure 9). Then, by Corollary [latex]3, \\, f^{\\prime}[\/latex] is an increasing function over [latex]I[\/latex]. Since [latex]f^{\\prime}(b)=0[\/latex], we conclude that for all [latex]x \\in I[\/latex], [latex]f^{\\prime}(x)&lt;0[\/latex] if [latex]x&lt;b[\/latex] and [latex]f^{\\prime}(x)&gt;0[\/latex] if [latex]x&gt;b[\/latex]. Therefore, by the first derivative test, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"550\"]<img id=\"21\" class=\"\" src=\"https:\/\/openstax.org\/apps\/archive\/20210421.141058\/resources\/b4c5748b2ddf90fa36c58b811e8bda2a1d0025e5\" alt=\"A function f(x) is graphed in the first quadrant with a and b marked on the x-axis. The function is vaguely sinusoidal, increasing first to x = a, then decreasing to x = b, and increasing again. At (a, f(a)), the tangent is marked, and it is noted that f\u2019(a) = 0 and f\u2019\u2019(a) &lt; 0. At (b, f(b)), the tangent is marked, and it is noted f\u2019(b) = 0 and f\u2019\u2019(b) &gt; 0.\" width=\"550\" height=\"307\" data-media-type=\"image\/jpeg\" \/> Figure 9. Consider a twice-differentiable function [latex]f[\/latex] such that [latex]f^{\\prime \\prime}[\/latex] is continuous. Since [latex]f^{\\prime}(a)=0[\/latex] and [latex]f^{\\prime \\prime}(a)&lt;0[\/latex], there is an interval [latex]I[\/latex] containing [latex]a[\/latex] such that for all [latex]x[\/latex] in [latex]I[\/latex], [latex]f[\/latex] is increasing if [latex]x&lt;a[\/latex] and [latex]f[\/latex] is decreasing if [latex]x&gt;a[\/latex]. As a result, [latex]f[\/latex] has a local maximum at [latex]x=a[\/latex]. Since [latex]f^{\\prime}(b)=0[\/latex] and [latex]f^{\\prime \\prime}(b)&gt;0[\/latex], there is an interval [latex]I[\/latex] containing [latex]b[\/latex] such that for all [latex]x[\/latex] in [latex]I[\/latex], [latex]f[\/latex] is decreasing if [latex]x&lt;b[\/latex] and [latex]f[\/latex] is increasing if [latex]x&gt;b[\/latex]. As a result, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex].[\/caption]\r\n<div id=\"fs-id1165043254223\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">Second Derivative Test<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165043254229\">Suppose [latex]f^{\\prime}(c)=0, \\, f^{\\prime \\prime}[\/latex] is continuous over an interval containing [latex]c[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1165042319133\">\r\n \t<li>If [latex]f^{\\prime \\prime}(c)&gt;0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime \\prime}(c)&lt;0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime \\prime}(c)=0[\/latex], then the test is inconclusive.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<p id=\"fs-id1165043180196\">Note that for case iii. when [latex]f^{\\prime \\prime}(c)=0[\/latex], then [latex]f[\/latex] may have a local maximum, local minimum, or neither at [latex]c[\/latex]. For example, the functions [latex]f(x)=x^3[\/latex], [latex]f(x)=x^4[\/latex], and [latex]f(x)=\u2212x^4[\/latex] all have critical points at [latex]x=0[\/latex]. In each case, the second derivative is zero at [latex]x=0[\/latex]. However, the function [latex]f(x)=x^4[\/latex] has a local minimum at [latex]x=0[\/latex] whereas the function [latex]f(x)=\u2212x^4[\/latex] has a local maximum at [latex]x=0[\/latex] and the function [latex]f(x)=x^3[\/latex] does not have a local extremum at [latex]x=0[\/latex].<\/p>\r\n<p id=\"fs-id1165043425387\">Let\u2019s now look at how to use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at a critical point [latex]c[\/latex] where [latex]f^{\\prime}(c)=0[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165042710827\" class=\"textbook exercises\">\r\n<h3>Example: Using the Second Derivative Test<\/h3>\r\nUse the second derivative to find the location of all local extrema for [latex]f(x)=x^5-5x^3[\/latex].\r\n<div id=\"fs-id1165042710829\" class=\"exercise\">[reveal-answer q=\"fs-id1165042320876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165042320876\"]\r\n<p id=\"fs-id1165042320876\">To apply the second derivative test, we first need to find critical points [latex]c[\/latex] where [latex]f^{\\prime}(c)=0[\/latex]. The derivative is [latex]f^{\\prime}(x)=5x^4-15x^2[\/latex]. Therefore, [latex]f^{\\prime}(x)=5x^4-15x^2=5x^2(x^2-3)=0[\/latex] when [latex]x=0,\\pm \\sqrt{3}[\/latex].<\/p>\r\n<p id=\"fs-id1165043431475\">To determine whether [latex]f[\/latex] has a local extrema at any of these points, we need to evaluate the sign of [latex]f^{\\prime \\prime}[\/latex] at these points. The second derivative is<\/p>\r\n\r\n<div id=\"fs-id1165042318723\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime \\prime}(x)=20x^3-30x=10x(2x^2-3)[\/latex].<\/div>\r\n<p id=\"fs-id1165042373916\">In the following table, we evaluate the second derivative at each of the critical points and use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at any of these points.<\/p>\r\n\r\n<table id=\"fs-id1165042705774\" class=\"unnumbered\" summary=\"This table has four rows and three columns. The first row is a header row, and it reads x, f\u2019\u2019(x), and Conclusion. After the header, the first column reads negative square root of 3, 0, and square root of 3. The second column reads negative 30 times the square root of 3, 0, and 30 times the square root of 3. The third column reads Local maxiumum, Second derivative test is inconclusive, and Local minimum.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f^{\\prime \\prime}(x)[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\u2212\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]-30\\sqrt{3}[\/latex]<\/td>\r\n<td>Local maximum<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<td>Second derivative test is inconclusive<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]30\\sqrt{3}[\/latex]<\/td>\r\n<td>Local minimum<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042476082\">By the second derivative test, we conclude that [latex]f[\/latex] has a local maximum at [latex]x=\u2212\\sqrt{3}[\/latex] and [latex]f[\/latex] has a local minimum at [latex]x=\\sqrt{3}[\/latex]. The second derivative test is inconclusive at [latex]x=0[\/latex]. To determine whether [latex]f[\/latex] has a local extrema at [latex]x=0[\/latex], we apply the first derivative test. To evaluate the sign of [latex]f^{\\prime}(x)=5x^2(x^2-3)[\/latex] for [latex]x \\in (\u2212\\sqrt{3},0)[\/latex] and [latex]x \\in (0,\\sqrt{3})[\/latex], let [latex]x=-1[\/latex] and [latex]x=1[\/latex] be the two test points. Since [latex]f^{\\prime}(-1)&lt;0[\/latex] and [latex]f^{\\prime}(1)&lt;0[\/latex], we conclude that [latex]f[\/latex] is decreasing on both intervals and, therefore, [latex]f[\/latex] does not have a local extrema at [latex]x=0[\/latex] as shown in the following graph.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"801\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210942\/CNX_Calc_Figure_04_05_007.jpg\" alt=\"The function f(x) = x5 \u2013 5x3 is graphed. The function increases to (negative square root of 3, 10), then decreases to an inflection point at 0, continues decreasing to (square root of 3, \u221210), and then increases.\" width=\"801\" height=\"408\" \/> Figure 10. The function [latex]f[\/latex] has a local maximum at [latex]x=\u2212\\sqrt{3}[\/latex] and a local minimum at [latex]x=\\sqrt{3}[\/latex][\/caption][\/hidden-answer]<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Using the Second Derivative Test.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/PBeo4ZJ-FGY?controls=0&amp;start=789&amp;end=1016&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\r\n<div id=\"1622857412.000500\" class=\"c-virtual_list__item\" tabindex=\"-1\" role=\"listitem\" aria-expanded=\"false\" data-qa=\"virtual-list-item\">\r\n<div class=\"c-message_kit__background p-message_pane_message__message c-message_kit__message\" role=\"document\" data-qa=\"message_container\" data-qa-unprocessed=\"false\" data-qa-placeholder=\"false\">\r\n<div class=\"c-message_kit__hover\" role=\"document\" data-qa-hover=\"true\">\r\n<div class=\"c-message_kit__actions c-message_kit__actions--above\">\r\n<div class=\"c-message_kit__gutter\">\r\n<div class=\"c-message_kit__gutter__right\" data-qa=\"message_content\">\r\n<div class=\"c-message_kit__blocks c-message_kit__blocks--rich_text\">\r\n<div class=\"c-message__message_blocks c-message__message_blocks--rich_text\">\r\n<div class=\"p-block_kit_renderer\" data-qa=\"block-kit-renderer\">\r\n<div class=\"p-block_kit_renderer__block_wrapper p-block_kit_renderer__block_wrapper--first\">\r\n<div class=\"p-rich_text_block\" dir=\"auto\">\r\n<div><\/div>\r\n[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 style=\"font-size: 1rem; orphans: 1; text-align: initial;\" href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.5DerivativesAndTheShapeOfAGraph789to1016_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.5 Derivatives and the Shape of a Graph\" here (opens in new window)<\/a>[\/hidden-answer]\r\n<div><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConsider the function [latex]f(x)=x^3-\\left(\\frac{3}{2}\\right)x^2-18x[\/latex]. The points [latex]c=3,-2[\/latex] satisfy [latex]f^{\\prime}(c)=0[\/latex]. Use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at those points.\r\n\r\n[reveal-answer q=\"707252\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"707252\"]\r\n\r\n[latex]f^{\\prime \\prime}(x)=6x-3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165043173990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043173990\"]\r\n\r\n[latex]f[\/latex] has a local maximum at -2 and a local minimum at 3.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]209260[\/ohm_question]\r\n\r\n<\/div>\r\nWe have now developed the tools we need to determine where a function is increasing and decreasing, as well as acquired an understanding of the basic shape of the graph. Next we discuss what happens to a function as [latex]x \\to \\pm \\infty[\/latex]. At that point, we have enough tools to provide accurate graphs of a large variety of functions.\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Explain the relationship between a function and its first and second derivatives<\/li>\n<li>State the second derivative test for local extrema<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165043423622\" class=\"bc-section section\">\n<p id=\"fs-id1165042621320\">The first derivative test provides an analytical tool for finding local extrema, but the second derivative can also be used to locate extreme values. Using the second derivative can sometimes be a simpler method than using the first derivative.<\/p>\n<p id=\"fs-id1165042621325\">We know that if a continuous function has a local extrema, it must occur at a critical point. However, a function need not have a local extrema at a critical point. Here we examine how the<strong> second derivative test<\/strong> can be used to determine whether a function has a local extremum at a critical point. Let [latex]f[\/latex] be a twice-differentiable function such that [latex]f^{\\prime}(a)=0[\/latex] and [latex]f^{\\prime \\prime}[\/latex] is continuous over an open interval [latex]I[\/latex] containing [latex]a[\/latex]. Suppose [latex]f^{\\prime \\prime}(a)<0[\/latex]. Since [latex]f^{\\prime \\prime}[\/latex] is continuous over [latex]I[\/latex], [latex]f^{\\prime \\prime}(x)<0[\/latex] for all [latex]x \\in I[\/latex] (Figure 9). Then, by Corollary 3, [latex]f^{\\prime}[\/latex] is a decreasing function over [latex]I[\/latex]. Since [latex]f^{\\prime}(a)=0[\/latex], we conclude that for all [latex]x \\in I, \\, f^{\\prime}(x)>0[\/latex] if [latex]x<a[\/latex] and [latex]f^{\\prime}(x)<0[\/latex] if [latex]x>a[\/latex]. Therefore, by the first derivative test, [latex]f[\/latex] has a local maximum at [latex]x=a[\/latex]. On the other hand, suppose there exists a point [latex]b[\/latex] such that [latex]f^{\\prime}(b)=0[\/latex] but [latex]f^{\\prime \\prime}(b)>0[\/latex]. Since [latex]f^{\\prime \\prime}[\/latex] is continuous over an open interval [latex]I[\/latex] containing [latex]b[\/latex], then [latex]f^{\\prime \\prime}(x)>0[\/latex] for all [latex]x \\in I[\/latex] (Figure 9). Then, by Corollary [latex]3, \\, f^{\\prime}[\/latex] is an increasing function over [latex]I[\/latex]. Since [latex]f^{\\prime}(b)=0[\/latex], we conclude that for all [latex]x \\in I[\/latex], [latex]f^{\\prime}(x)<0[\/latex] if [latex]x<b[\/latex] and [latex]f^{\\prime}(x)>0[\/latex] if [latex]x>b[\/latex]. Therefore, by the first derivative test, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex].<\/p>\n<div style=\"width: 560px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" id=\"21\" class=\"\" src=\"https:\/\/openstax.org\/apps\/archive\/20210421.141058\/resources\/b4c5748b2ddf90fa36c58b811e8bda2a1d0025e5\" alt=\"A function f(x) is graphed in the first quadrant with a and b marked on the x-axis. The function is vaguely sinusoidal, increasing first to x = a, then decreasing to x = b, and increasing again. At (a, f(a)), the tangent is marked, and it is noted that f\u2019(a) = 0 and f\u2019\u2019(a) &lt; 0. At (b, f(b)), the tangent is marked, and it is noted f\u2019(b) = 0 and f\u2019\u2019(b) &gt; 0.\" width=\"550\" height=\"307\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9. Consider a twice-differentiable function [latex]f[\/latex] such that [latex]f^{\\prime \\prime}[\/latex] is continuous. Since [latex]f^{\\prime}(a)=0[\/latex] and [latex]f^{\\prime \\prime}(a)&lt;0[\/latex], there is an interval [latex]I[\/latex] containing [latex]a[\/latex] such that for all [latex]x[\/latex] in [latex]I[\/latex], [latex]f[\/latex] is increasing if [latex]x&lt;a[\/latex] and [latex]f[\/latex] is decreasing if [latex]x&gt;a[\/latex]. As a result, [latex]f[\/latex] has a local maximum at [latex]x=a[\/latex]. Since [latex]f^{\\prime}(b)=0[\/latex] and [latex]f^{\\prime \\prime}(b)&gt;0[\/latex], there is an interval [latex]I[\/latex] containing [latex]b[\/latex] such that for all [latex]x[\/latex] in [latex]I[\/latex], [latex]f[\/latex] is decreasing if [latex]x&lt;b[\/latex] and [latex]f[\/latex] is increasing if [latex]x&gt;b[\/latex]. As a result, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165043254223\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">Second Derivative Test<\/h3>\n<hr \/>\n<p id=\"fs-id1165043254229\">Suppose [latex]f^{\\prime}(c)=0, \\, f^{\\prime \\prime}[\/latex] is continuous over an interval containing [latex]c[\/latex].<\/p>\n<ol id=\"fs-id1165042319133\">\n<li>If [latex]f^{\\prime \\prime}(c)>0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(c)<0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(c)=0[\/latex], then the test is inconclusive.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1165043180196\">Note that for case iii. when [latex]f^{\\prime \\prime}(c)=0[\/latex], then [latex]f[\/latex] may have a local maximum, local minimum, or neither at [latex]c[\/latex]. For example, the functions [latex]f(x)=x^3[\/latex], [latex]f(x)=x^4[\/latex], and [latex]f(x)=\u2212x^4[\/latex] all have critical points at [latex]x=0[\/latex]. In each case, the second derivative is zero at [latex]x=0[\/latex]. However, the function [latex]f(x)=x^4[\/latex] has a local minimum at [latex]x=0[\/latex] whereas the function [latex]f(x)=\u2212x^4[\/latex] has a local maximum at [latex]x=0[\/latex] and the function [latex]f(x)=x^3[\/latex] does not have a local extremum at [latex]x=0[\/latex].<\/p>\n<p id=\"fs-id1165043425387\">Let\u2019s now look at how to use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at a critical point [latex]c[\/latex] where [latex]f^{\\prime}(c)=0[\/latex].<\/p>\n<div id=\"fs-id1165042710827\" class=\"textbook exercises\">\n<h3>Example: Using the Second Derivative Test<\/h3>\n<p>Use the second derivative to find the location of all local extrema for [latex]f(x)=x^5-5x^3[\/latex].<\/p>\n<div id=\"fs-id1165042710829\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165042320876\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165042320876\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042320876\">To apply the second derivative test, we first need to find critical points [latex]c[\/latex] where [latex]f^{\\prime}(c)=0[\/latex]. The derivative is [latex]f^{\\prime}(x)=5x^4-15x^2[\/latex]. Therefore, [latex]f^{\\prime}(x)=5x^4-15x^2=5x^2(x^2-3)=0[\/latex] when [latex]x=0,\\pm \\sqrt{3}[\/latex].<\/p>\n<p id=\"fs-id1165043431475\">To determine whether [latex]f[\/latex] has a local extrema at any of these points, we need to evaluate the sign of [latex]f^{\\prime \\prime}[\/latex] at these points. The second derivative is<\/p>\n<div id=\"fs-id1165042318723\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime \\prime}(x)=20x^3-30x=10x(2x^2-3)[\/latex].<\/div>\n<p id=\"fs-id1165042373916\">In the following table, we evaluate the second derivative at each of the critical points and use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at any of these points.<\/p>\n<table id=\"fs-id1165042705774\" class=\"unnumbered\" summary=\"This table has four rows and three columns. The first row is a header row, and it reads x, f\u2019\u2019(x), and Conclusion. After the header, the first column reads negative square root of 3, 0, and square root of 3. The second column reads negative 30 times the square root of 3, 0, and 30 times the square root of 3. The third column reads Local maxiumum, Second derivative test is inconclusive, and Local minimum.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f^{\\prime \\prime}(x)[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\u2212\\sqrt{3}[\/latex]<\/td>\n<td>[latex]-30\\sqrt{3}[\/latex]<\/td>\n<td>Local maximum<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>0<\/td>\n<td>0<\/td>\n<td>Second derivative test is inconclusive<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>[latex]30\\sqrt{3}[\/latex]<\/td>\n<td>Local minimum<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042476082\">By the second derivative test, we conclude that [latex]f[\/latex] has a local maximum at [latex]x=\u2212\\sqrt{3}[\/latex] and [latex]f[\/latex] has a local minimum at [latex]x=\\sqrt{3}[\/latex]. The second derivative test is inconclusive at [latex]x=0[\/latex]. To determine whether [latex]f[\/latex] has a local extrema at [latex]x=0[\/latex], we apply the first derivative test. To evaluate the sign of [latex]f^{\\prime}(x)=5x^2(x^2-3)[\/latex] for [latex]x \\in (\u2212\\sqrt{3},0)[\/latex] and [latex]x \\in (0,\\sqrt{3})[\/latex], let [latex]x=-1[\/latex] and [latex]x=1[\/latex] be the two test points. Since [latex]f^{\\prime}(-1)<0[\/latex] and [latex]f^{\\prime}(1)<0[\/latex], we conclude that [latex]f[\/latex] is decreasing on both intervals and, therefore, [latex]f[\/latex] does not have a local extrema at [latex]x=0[\/latex] as shown in the following graph.<\/p>\n<div style=\"width: 811px\" 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\/11210942\/CNX_Calc_Figure_04_05_007.jpg\" alt=\"The function f(x) = x5 \u2013 5x3 is graphed. The function increases to (negative square root of 3, 10), then decreases to an inflection point at 0, continues decreasing to (square root of 3, \u221210), and then increases.\" width=\"801\" height=\"408\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 10. The function [latex]f[\/latex] has a local maximum at [latex]x=\u2212\\sqrt{3}[\/latex] and a local minimum at [latex]x=\\sqrt{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Using the Second Derivative Test.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/PBeo4ZJ-FGY?controls=0&amp;start=789&amp;end=1016&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<div id=\"1622857412.000500\" class=\"c-virtual_list__item\" tabindex=\"-1\" role=\"listitem\" aria-expanded=\"false\" data-qa=\"virtual-list-item\">\n<div class=\"c-message_kit__background p-message_pane_message__message c-message_kit__message\" role=\"document\" data-qa=\"message_container\" data-qa-unprocessed=\"false\" data-qa-placeholder=\"false\">\n<div class=\"c-message_kit__hover\" role=\"document\" data-qa-hover=\"true\">\n<div class=\"c-message_kit__actions c-message_kit__actions--above\">\n<div class=\"c-message_kit__gutter\">\n<div class=\"c-message_kit__gutter__right\" data-qa=\"message_content\">\n<div class=\"c-message_kit__blocks c-message_kit__blocks--rich_text\">\n<div class=\"c-message__message_blocks c-message__message_blocks--rich_text\">\n<div class=\"p-block_kit_renderer\" data-qa=\"block-kit-renderer\">\n<div class=\"p-block_kit_renderer__block_wrapper p-block_kit_renderer__block_wrapper--first\">\n<div class=\"p-rich_text_block\" dir=\"auto\">\n<div><\/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 style=\"font-size: 1rem; orphans: 1; text-align: initial;\" href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.5DerivativesAndTheShapeOfAGraph789to1016_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.5 Derivatives and the Shape of a Graph&#8221; here (opens in new window)<\/a><\/div>\n<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Consider the function [latex]f(x)=x^3-\\left(\\frac{3}{2}\\right)x^2-18x[\/latex]. The points [latex]c=3,-2[\/latex] satisfy [latex]f^{\\prime}(c)=0[\/latex]. Use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at those points.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q707252\">Hint<\/span><\/p>\n<div id=\"q707252\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f^{\\prime \\prime}(x)=6x-3[\/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-id1165043173990\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043173990\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f[\/latex] has a local maximum at -2 and a local minimum at 3.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm209260\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=209260&theme=oea&iframe_resize_id=ohm209260&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>We have now developed the tools we need to determine where a function is increasing and decreasing, as well as acquired an understanding of the basic shape of the graph. Next we discuss what happens to a function as [latex]x \\to \\pm \\infty[\/latex]. At that point, we have enough tools to provide accurate graphs of a large variety of functions.<\/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-404\">\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>4.5 Derivatives and the Shape of a Graph. <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":17,"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\":\"4.5 Derivatives and the Shape of a Graph\",\"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-404","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/404","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":19,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/404\/revisions"}],"predecessor-version":[{"id":4837,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/404\/revisions\/4837"}],"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\/404\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=404"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=404"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=404"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=404"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}