{"id":135,"date":"2023-06-21T13:22:36","date_gmt":"2023-06-21T13:22:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/behaviors-of-functions\/"},"modified":"2023-08-24T05:25:41","modified_gmt":"2023-08-24T05:25:41","slug":"behaviors-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/behaviors-of-functions\/","title":{"raw":"\u25aa   Behaviors of Functions","rendered":"\u25aa   Behaviors of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine where a function is increasing, decreasing, or constant<\/li>\r\n \t<li>Find local extrema of a function from a graph<\/li>\r\n \t<li>Describe behavior of the toolkit functions<\/li>\r\n<\/ul>\r\n<\/div>\r\nAs part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. The graph below\u00a0shows examples of increasing and decreasing intervals on a function.\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\/896\/2016\/10\/18194750\/CNX_Precalc_Figure_01_03_0042.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.\" width=\"487\" height=\"518\" \/> The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}-\\text{2}\\right){{\\cup }^{\\text{ }}}^{\\text{ }}\\left(2,\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}2\\right)[\/latex].[\/caption]This video further explains how to find where a function is increasing or decreasing.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=78b4HOMVcKM\r\n\r\nWhile some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the output where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a <strong>local maximum<\/strong>. If a function has more than one, we say it has local maxima. Similarly, a value of the output where a function changes from decreasing to increasing as the input variable increases is called a <strong>local minimum<\/strong>. The plural form is \"local minima.\" Together, local maxima and minima are called <strong>local extrema<\/strong>, or local extreme values, of the function. (The singular form is \"extremum.\") Often, the term <em>local<\/em> is replaced by the term <em>relative<\/em>. In this text, we will use the term <em>local<\/em>.\r\n\r\nA function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of <em>local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.\r\n\r\nFor the function below, the local maximum is 16, and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194752\/CNX_Precalc_Figure_01_03_0142.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum. The local maximum is 16 and occurs at x = negative 2. This is the point negative 2, 16. The local minimum is negative 16 and occurs at x = 2. This is the point 2, negative 16.\" width=\"731\" height=\"467\" \/>\r\n\r\nTo locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. The graph below\u00a0illustrates these ideas for a local maximum.\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\/896\/2016\/10\/18194754\/CNX_Precalc_Figure_01_03_0052.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.\" width=\"487\" height=\"295\" \/> Definition of a local maximum.[\/caption]\r\n\r\nThese observations lead us to a formal definition of local extrema.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Local Minima and Local Maxima<\/h3>\r\nA function [latex]f[\/latex] is an <strong>increasing function<\/strong> on an open interval if [latex]f\\left(b\\right)&gt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex].\r\n\r\nA function [latex]f[\/latex] is a <strong>decreasing function<\/strong> on an open interval if [latex]f\\left(b\\right)&lt;f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b&gt;a[\/latex].\r\n\r\nA function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a&lt;b&lt;c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\le f\\left(b\\right)[\/latex]. Likewise, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a&lt;b&lt;c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\ge f\\left(b\\right)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Increasing and Decreasing Intervals on a Graph<\/h3>\r\nGiven the function [latex]p\\left(t\\right)[\/latex] in the graph below, identify the intervals on which the function appears to be increasing.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194756\/CNX_Precalc_Figure_01_03_0062.jpg\" alt=\"Graph of a polynomial. As x gets large in the negative direction, the outputs of the function get large in the positive direction. As inputs approach 1, then the function value approaches a minimum of negative one. As x approaches 3, the values increase again and between 3 and 4 decrease one last time. As x gets large in the positive direction, the function values increase without bound.\" width=\"487\" height=\"295\" \/>\r\n[reveal-answer q=\"927495\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"927495\"]\r\n\r\nWe see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on.\r\n\r\nIn <strong>interval notation<\/strong>, we would say the function appears to be increasing on the interval [latex](1,3)[\/latex] and the interval [latex]\\left(4,\\infty \\right)[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nNotice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[\/latex] , [latex]t=3[\/latex] , and [latex]t=4[\/latex] . These points are the local extrema (two minima and a maximum).\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"420\"]4084[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Tip for success: increasing\/decreasing behavior<\/h3>\r\nThe behavior of the function values of the graph of a function is read over the x-axis, from left to right. That is,\r\n<ul>\r\n \t<li style=\"text-align: left;\">a function is said to be increasing if its function values increase as x increases;<\/li>\r\n \t<li style=\"text-align: left;\">a function is said to be decreasing if its function values decrease as x increases.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Local Extrema from a Graph<\/h3>\r\nGraph the function [latex]f\\left(x\\right)=\\dfrac{2}{x}+\\dfrac{x}{3}[\/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.\r\n\r\n[reveal-answer q=\"818075\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"818075\"]\r\n\r\nUsing technology, we find that the graph of the function looks like that below. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3[\/latex], and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194759\/CNX_Precalc_Figure_01_03_0072.jpg\" alt=\"Graph of a reciprocal function.\" width=\"487\" height=\"368\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nMost graphing calculators and graphing utilities can estimate the location of maxima and minima. The graph below\u00a0provides screen images from two different technologies, showing the estimate for the local maximum and minimum.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194802\/CNX_Precalc_Figure_01_03_008ab2.jpg\" alt=\"Graph of the reciprocal function on a graphing calculator.\" width=\"975\" height=\"376\" \/>\r\n\r\nBased on these estimates, the function is increasing on the interval [latex]\\left(-\\infty ,-{2.449}\\right)[\/latex] and [latex]\\left(2.449\\text{,}\\infty \\right)[\/latex]. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact locations of the extrema are at [latex]\\pm \\sqrt{6}[\/latex], but determining this requires calculus.)\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success: reading extrema<\/h3>\r\nRecall that points on the graph of a function are ordered pairs in the form of\r\n<p style=\"text-align: center;\">[latex]\\left(\\text{input, output}\\right) \\quad = \\quad \\left(x, f(x)\\right)[\/latex].<\/p>\r\nIf a function's graph has a local minimum or maximum at some point [latex]\\left(x, f(x)\\right)[\/latex], we say\r\n<p style=\"text-align: center;\">\"the extrema\u00a0<em>occurs at <\/em>[latex]x[\/latex], and that the minimum or maximum\u00a0<em>is\u00a0<\/em>[latex]f(x)[\/latex].\"<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGraph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.\r\n\r\n[reveal-answer q=\"466198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"466198\"]\r\n\r\nThe local maximum appears to occur at [latex]\\left(-1,28\\right)[\/latex], and the local minimum occurs at [latex]\\left(5,-80\\right)[\/latex]. The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right)[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-6728\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08193610\/Screen-Shot-2019-07-08-at-12.35.38-PM.png\" alt=\"Graph of a polynomial with a maximum at (-1,28) and a minimum at (5,-80).\" width=\"490\" height=\"549\" \/>\r\n\r\n<span id=\"fs-id1165134043615\">\u00a0<\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"350\"]32572[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Local Maxima and Minima from a Graph<\/h3>\r\nFor the function [latex]f[\/latex] whose graph is shown below, find all local maxima and minima.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194804\/CNX_Precalc_Figure_01_03_0112.jpg\" alt=\"Graph of a polynomial. The line curves down to x = negative 2 and up to x = 1.\" width=\"487\" height=\"368\" \/>\r\n\r\n[reveal-answer q=\"523190\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"523190\"]\r\n\r\nObserve the graph of [latex]f[\/latex]. The graph attains a local maximum at [latex]x=1[\/latex] because it is the highest point in an open interval around [latex]x=1[\/latex]. The local maximum is the [latex]y[\/latex] -coordinate at [latex]x=1[\/latex], which is [latex]2[\/latex].\r\n\r\nThe graph attains a local minimum at [latex]\\text{ }x=-1\\text{ }[\/latex] because it is the lowest point in an open interval around [latex]x=-1[\/latex]. The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1[\/latex], which is [latex]-2[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"520\"]32571[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success: toolkit functions<\/h3>\r\nThe toolkit functions continue to appear throughout the course. Have you memorized them yet? Would you be able to sketch a quick graph of each from its equation?\r\n\r\n<\/div>\r\n<h2>Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h2>\r\nWe will now return to our toolkit functions and discuss their graphical behavior in the table\u00a0below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Function<\/th>\r\n<th>Increasing\/Decreasing<\/th>\r\n<th>Example<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Constant Function[latex]f\\left(x\\right)={c}[\/latex]<\/td>\r\n<td>Neither increasing nor decreasing<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\"><img class=\"alignnone wp-image-12510 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194806\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\" alt=\"\" width=\"143\" height=\"146\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Identity Function[latex]f\\left(x\\right)={x}[\/latex]<\/td>\r\n<td>\u00a0Increasing<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\"><img class=\"alignnone wp-image-12511 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194807\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\" alt=\"\" width=\"143\" height=\"147\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Quadratic Function[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\r\n<td>Increasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]\r\n\r\nMinimum at [latex]x=0[\/latex]<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\"><img class=\"alignnone size-full wp-image-12512\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194808\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\" alt=\"\" width=\"138\" height=\"143\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cubic Function[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\r\n<td>Increasing<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\"><img class=\"alignnone wp-image-12513 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194809\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\" alt=\"\" width=\"137\" height=\"145\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0Reciprocal[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\r\n<td>Decreasing [latex]\\left(-\\infty,0\\right)\\cup\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\"><img class=\"alignnone wp-image-12514 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194810\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\" alt=\"\" width=\"138\" height=\"146\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reciprocal Squared[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\r\n<td>Increasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]Decreasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\"><img class=\"alignnone size-full wp-image-12515\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194812\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\" alt=\"\" width=\"134\" height=\"145\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Cube Root[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\r\n<td>Increasing<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\"><img class=\"alignnone size-full wp-image-12516\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194813\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\" alt=\"Screen Shot 2015-08-20 at 8.53.26 AM\" width=\"140\" height=\"147\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Square Root[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\r\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\"><img class=\"alignnone size-full wp-image-12517\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194814\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\" alt=\"\" width=\"138\" height=\"142\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Absolute Value[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\r\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]\r\n\r\nMinimum at [latex]x=0[\/latex]<\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\"><img class=\"alignnone wp-image-12518 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194814\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\" alt=\"\" width=\"135\" height=\"143\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Use a graph to locate the absolute maximum and absolute minimum of a function<\/h2>\r\nThere is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the <strong>absolute maximum <\/strong>and<strong> absolute minimum<\/strong>, respectively.\r\n\r\nTo locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194816\/CNX_Precalc_Figure_01_03_0152.jpg\" alt=\"Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).\" width=\"487\" height=\"323\" \/>\r\n\r\nNot every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Absolute Maxima and Minima<\/h3>\r\nThe <strong>absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].\r\n\r\nThe <strong>absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Absolute Maxima and Minima from a Graph<\/h3>\r\nFor the function [latex]f[\/latex] shown below, find all absolute maxima and minima.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194818\/CNX_Precalc_Figure_01_03_0132.jpg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"403\" \/>\r\n[reveal-answer q=\"461473\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"461473\"]\r\n\r\nObserve the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the <em>y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].\r\n\r\nThe graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the <em>y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine where a function is increasing, decreasing, or constant<\/li>\n<li>Find local extrema of a function from a graph<\/li>\n<li>Describe behavior of the toolkit functions<\/li>\n<\/ul>\n<\/div>\n<p>As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways. We say that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. The graph below\u00a0shows examples of increasing and decreasing intervals on a function.<\/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\/896\/2016\/10\/18194750\/CNX_Precalc_Figure_01_03_0042.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum.\" width=\"487\" height=\"518\" \/><\/p>\n<p class=\"wp-caption-text\">The function [latex]f\\left(x\\right)={x}^{3}-12x[\/latex] is increasing on [latex]\\left(-\\infty \\text{,}-\\text{2}\\right){{\\cup }^{\\text{ }}}^{\\text{ }}\\left(2,\\infty \\right)[\/latex] and is decreasing on [latex]\\left(-2\\text{,}2\\right)[\/latex].<\/p>\n<\/div>\n<p>This video further explains how to find where a function is increasing or decreasing.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine Where a Function is Increasing and Decreasing\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/78b4HOMVcKM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>While some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the output where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a <strong>local maximum<\/strong>. If a function has more than one, we say it has local maxima. Similarly, a value of the output where a function changes from decreasing to increasing as the input variable increases is called a <strong>local minimum<\/strong>. The plural form is &#8220;local minima.&#8221; Together, local maxima and minima are called <strong>local extrema<\/strong>, or local extreme values, of the function. (The singular form is &#8220;extremum.&#8221;) Often, the term <em>local<\/em> is replaced by the term <em>relative<\/em>. In this text, we will use the term <em>local<\/em>.<\/p>\n<p>A function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at extrema. Note that we have to speak of <em>local<\/em> extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function\u2019s entire domain.<\/p>\n<p>For the function below, the local maximum is 16, and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194752\/CNX_Precalc_Figure_01_03_0142.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum and minimum. The local maximum is 16 and occurs at x = negative 2. This is the point negative 2, 16. The local minimum is negative 16 and occurs at x = 2. This is the point 2, negative 16.\" width=\"731\" height=\"467\" \/><\/p>\n<p>To locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local minimum than at neighboring points. The graph below\u00a0illustrates these ideas for a local maximum.<\/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\/896\/2016\/10\/18194754\/CNX_Precalc_Figure_01_03_0052.jpg\" alt=\"Graph of a polynomial that shows the increasing and decreasing intervals and local maximum.\" width=\"487\" height=\"295\" \/><\/p>\n<p class=\"wp-caption-text\">Definition of a local maximum.<\/p>\n<\/div>\n<p>These observations lead us to a formal definition of local extrema.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Local Minima and Local Maxima<\/h3>\n<p>A function [latex]f[\/latex] is an <strong>increasing function<\/strong> on an open interval if [latex]f\\left(b\\right)>f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex].<\/p>\n<p>A function [latex]f[\/latex] is a <strong>decreasing function<\/strong> on an open interval if [latex]f\\left(b\\right)<f\\left(a\\right)[\/latex] for any two input values [latex]a[\/latex] and [latex]b[\/latex] in the given interval where [latex]b>a[\/latex].<\/p>\n<p>A function [latex]f[\/latex] has a local maximum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a<b<c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\le f\\left(b\\right)[\/latex]. Likewise, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex] if there exists an interval [latex]\\left(a,c\\right)[\/latex] with [latex]a<b<c[\/latex] such that, for any [latex]x[\/latex] in the interval [latex]\\left(a,c\\right)[\/latex], [latex]f\\left(x\\right)\\ge f\\left(b\\right)[\/latex].\n\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Increasing and Decreasing Intervals on a Graph<\/h3>\n<p>Given the function [latex]p\\left(t\\right)[\/latex] in the graph below, identify the intervals on which the function appears to be increasing.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194756\/CNX_Precalc_Figure_01_03_0062.jpg\" alt=\"Graph of a polynomial. As x gets large in the negative direction, the outputs of the function get large in the positive direction. As inputs approach 1, then the function value approaches a minimum of negative one. As x approaches 3, the values increase again and between 3 and 4 decrease one last time. As x gets large in the positive direction, the function values increase without bound.\" width=\"487\" height=\"295\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q927495\">Show Solution<\/span><\/p>\n<div id=\"q927495\" class=\"hidden-answer\" style=\"display: none\">\n<p>We see that the function is not constant on any interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from [latex]t=1[\/latex] to [latex]t=3[\/latex] and from [latex]t=4[\/latex] on.<\/p>\n<p>In <strong>interval notation<\/strong>, we would say the function appears to be increasing on the interval [latex](1,3)[\/latex] and the interval [latex]\\left(4,\\infty \\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice in this example that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at [latex]t=1[\/latex] , [latex]t=3[\/latex] , and [latex]t=4[\/latex] . These points are the local extrema (two minima and a maximum).<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm4084\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4084&theme=oea&iframe_resize_id=ohm4084&show_question_numbers\" width=\"100%\" height=\"420\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Tip for success: increasing\/decreasing behavior<\/h3>\n<p>The behavior of the function values of the graph of a function is read over the x-axis, from left to right. That is,<\/p>\n<ul>\n<li style=\"text-align: left;\">a function is said to be increasing if its function values increase as x increases;<\/li>\n<li style=\"text-align: left;\">a function is said to be decreasing if its function values decrease as x increases.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Local Extrema from a Graph<\/h3>\n<p>Graph the function [latex]f\\left(x\\right)=\\dfrac{2}{x}+\\dfrac{x}{3}[\/latex]. Then use the graph to estimate the local extrema of the function and to determine the intervals on which the function is increasing.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q818075\">Show Solution<\/span><\/p>\n<div id=\"q818075\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using technology, we find that the graph of the function looks like that below. It appears there is a low point, or local minimum, between [latex]x=2[\/latex] and [latex]x=3[\/latex], and a mirror-image high point, or local maximum, somewhere between [latex]x=-3[\/latex] and [latex]x=-2[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194759\/CNX_Precalc_Figure_01_03_0072.jpg\" alt=\"Graph of a reciprocal function.\" width=\"487\" height=\"368\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Most graphing calculators and graphing utilities can estimate the location of maxima and minima. The graph below\u00a0provides screen images from two different technologies, showing the estimate for the local maximum and minimum.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194802\/CNX_Precalc_Figure_01_03_008ab2.jpg\" alt=\"Graph of the reciprocal function on a graphing calculator.\" width=\"975\" height=\"376\" \/><\/p>\n<p>Based on these estimates, the function is increasing on the interval [latex]\\left(-\\infty ,-{2.449}\\right)[\/latex] and [latex]\\left(2.449\\text{,}\\infty \\right)[\/latex]. Notice that, while we expect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing approximation algorithms used by each. (The exact locations of the extrema are at [latex]\\pm \\sqrt{6}[\/latex], but determining this requires calculus.)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success: reading extrema<\/h3>\n<p>Recall that points on the graph of a function are ordered pairs in the form of<\/p>\n<p style=\"text-align: center;\">[latex]\\left(\\text{input, output}\\right) \\quad = \\quad \\left(x, f(x)\\right)[\/latex].<\/p>\n<p>If a function&#8217;s graph has a local minimum or maximum at some point [latex]\\left(x, f(x)\\right)[\/latex], we say<\/p>\n<p style=\"text-align: center;\">&#8220;the extrema\u00a0<em>occurs at <\/em>[latex]x[\/latex], and that the minimum or maximum\u00a0<em>is\u00a0<\/em>[latex]f(x)[\/latex].&#8221;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Graph the function [latex]f\\left(x\\right)={x}^{3}-6{x}^{2}-15x+20[\/latex] to estimate the local extrema of the function. Use these to determine the intervals on which the function is increasing and decreasing.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q466198\">Show Solution<\/span><\/p>\n<div id=\"q466198\" class=\"hidden-answer\" style=\"display: none\">\n<p>The local maximum appears to occur at [latex]\\left(-1,28\\right)[\/latex], and the local minimum occurs at [latex]\\left(5,-80\\right)[\/latex]. The function is increasing on [latex]\\left(-\\infty ,-1\\right)\\cup \\left(5,\\infty \\right)[\/latex] and decreasing on [latex]\\left(-1,5\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6728\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/10\/08193610\/Screen-Shot-2019-07-08-at-12.35.38-PM.png\" alt=\"Graph of a polynomial with a maximum at (-1,28) and a minimum at (5,-80).\" width=\"490\" height=\"549\" \/><\/p>\n<p><span id=\"fs-id1165134043615\">\u00a0<\/span><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm32572\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=32572&theme=oea&iframe_resize_id=ohm32572&show_question_numbers\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Local Maxima and Minima from a Graph<\/h3>\n<p>For the function [latex]f[\/latex] whose graph is shown below, find all local maxima and minima.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194804\/CNX_Precalc_Figure_01_03_0112.jpg\" alt=\"Graph of a polynomial. The line curves down to x = negative 2 and up to x = 1.\" width=\"487\" height=\"368\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q523190\">Show Solution<\/span><\/p>\n<div id=\"q523190\" class=\"hidden-answer\" style=\"display: none\">\n<p>Observe the graph of [latex]f[\/latex]. The graph attains a local maximum at [latex]x=1[\/latex] because it is the highest point in an open interval around [latex]x=1[\/latex]. The local maximum is the [latex]y[\/latex] -coordinate at [latex]x=1[\/latex], which is [latex]2[\/latex].<\/p>\n<p>The graph attains a local minimum at [latex]\\text{ }x=-1\\text{ }[\/latex] because it is the lowest point in an open interval around [latex]x=-1[\/latex]. The local minimum is the <em>y<\/em>-coordinate at [latex]x=-1[\/latex], which is [latex]-2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm32571\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=32571&theme=oea&iframe_resize_id=ohm32571&show_question_numbers\" width=\"100%\" height=\"520\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success: toolkit functions<\/h3>\n<p>The toolkit functions continue to appear throughout the course. Have you memorized them yet? Would you be able to sketch a quick graph of each from its equation?<\/p>\n<\/div>\n<h2>Analyzing the Toolkit Functions for Increasing or Decreasing Intervals<\/h2>\n<p>We will now return to our toolkit functions and discuss their graphical behavior in the table\u00a0below.<\/p>\n<table>\n<thead>\n<tr>\n<th>Function<\/th>\n<th>Increasing\/Decreasing<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Constant Function[latex]f\\left(x\\right)={c}[\/latex]<\/td>\n<td>Neither increasing nor decreasing<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12510 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194806\/Screen-Shot-2015-08-20-at-8.52.37-AM.png\" alt=\"\" width=\"143\" height=\"146\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Identity Function[latex]f\\left(x\\right)={x}[\/latex]<\/td>\n<td>\u00a0Increasing<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12511 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194807\/Screen-Shot-2015-08-20-at-8.52.47-AM.png\" alt=\"\" width=\"143\" height=\"147\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Quadratic Function[latex]f\\left(x\\right)={x}^{2}[\/latex]<\/td>\n<td>Increasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]<\/p>\n<p>Minimum at [latex]x=0[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12512\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194808\/Screen-Shot-2015-08-20-at-8.52.54-AM.png\" alt=\"\" width=\"138\" height=\"143\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Cubic Function[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/td>\n<td>Increasing<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12513 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194809\/Screen-Shot-2015-08-20-at-8.53.02-AM.png\" alt=\"\" width=\"137\" height=\"145\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>\u00a0Reciprocal[latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]<\/td>\n<td>Decreasing [latex]\\left(-\\infty,0\\right)\\cup\\left(0,\\infty\\right)[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12514 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194810\/Screen-Shot-2015-08-20-at-8.53.09-AM.png\" alt=\"\" width=\"138\" height=\"146\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Reciprocal Squared[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]<\/td>\n<td>Increasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]Decreasing on\u00a0[latex]\\left(0,\\infty\\right)[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12515\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194812\/Screen-Shot-2015-08-20-at-8.53.16-AM.png\" alt=\"\" width=\"134\" height=\"145\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Cube Root[latex]f\\left(x\\right)=\\sqrt[3]{x}[\/latex]<\/td>\n<td>Increasing<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12516\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194813\/Screen-Shot-2015-08-20-at-8.53.26-AM.png\" alt=\"Screen Shot 2015-08-20 at 8.53.26 AM\" width=\"140\" height=\"147\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Square Root[latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/td>\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12517\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194814\/Screen-Shot-2015-08-20-at-8.53.33-AM.png\" alt=\"\" width=\"138\" height=\"142\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td>Absolute Value[latex]f\\left(x\\right)=|x|[\/latex]<\/td>\n<td>Increasing on [latex]\\left(0,\\infty\\right)[\/latex]Decreasing on\u00a0[latex]\\left(-\\infty,0\\right)[\/latex]<\/p>\n<p>Minimum at [latex]x=0[\/latex]<\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-12518 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194814\/Screen-Shot-2015-08-20-at-8.53.40-AM.png\" alt=\"\" width=\"135\" height=\"143\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Use a graph to locate the absolute maximum and absolute minimum of a function<\/h2>\n<p>There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The [latex]y\\text{-}[\/latex] coordinates (output) at the highest and lowest points are called the <strong>absolute maximum <\/strong>and<strong> absolute minimum<\/strong>, respectively.<\/p>\n<p>To locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194816\/CNX_Precalc_Figure_01_03_0152.jpg\" alt=\"Graph of a segment of a parabola with an absolute minimum at (0, -2) and absolute maximum at (2, 2).\" width=\"487\" height=\"323\" \/><\/p>\n<p>Not every function has an absolute maximum or minimum value. The toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex] is one such function.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Absolute Maxima and Minima<\/h3>\n<p>The <strong>absolute maximum<\/strong> of [latex]f[\/latex] at [latex]x=c[\/latex] is [latex]f\\left(c\\right)[\/latex] where [latex]f\\left(c\\right)\\ge f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n<p>The <strong>absolute minimum<\/strong> of [latex]f[\/latex] at [latex]x=d[\/latex] is [latex]f\\left(d\\right)[\/latex] where [latex]f\\left(d\\right)\\le f\\left(x\\right)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Absolute Maxima and Minima from a Graph<\/h3>\n<p>For the function [latex]f[\/latex] shown below, find all absolute maxima and minima.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18194818\/CNX_Precalc_Figure_01_03_0132.jpg\" alt=\"Graph of a polynomial.\" width=\"487\" height=\"403\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q461473\">Show Solution<\/span><\/p>\n<div id=\"q461473\" class=\"hidden-answer\" style=\"display: none\">\n<p>Observe the graph of [latex]f[\/latex]. The graph attains an absolute maximum in two locations, [latex]x=-2[\/latex] and [latex]x=2[\/latex], because at these locations, the graph attains its highest point on the domain of the function. The absolute maximum is the <em>y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].<\/p>\n<p>The graph attains an absolute minimum at [latex]x=3[\/latex], because it is the lowest point on the domain of the function\u2019s graph. The absolute minimum is the <em>y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].<\/p>\n<\/div>\n<\/div>\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-135\">\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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 4084. <strong>Authored by<\/strong>: Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 32572, 32571. <strong>Authored by<\/strong>: Smart,Jim. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/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":395986,"menu_order":24,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 4084\",\"author\":\"Lippman,David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License 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