{"id":599,"date":"2025-07-14T16:17:56","date_gmt":"2025-07-14T16:17:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=599"},"modified":"2026-01-07T21:02:09","modified_gmt":"2026-01-07T21:02:09","slug":"rates-of-change-and-behavior-of-graphs-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rates-of-change-and-behavior-of-graphs-learn-it-3\/","title":{"raw":"Rates of Change and Behavior of Graphs: Learn It 3","rendered":"Rates of Change and Behavior of Graphs: Learn It 3"},"content":{"raw":"

Local and Absolute Extrema<\/h2>\r\n
\r\n

local minima and local maxima (extrema)<\/h3>\r\n
    \r\n \t
  • A function [latex]f[\/latex] is an 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].<\/li>\r\n \t
  • A function [latex]f[\/latex] is a 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].<\/li>\r\n<\/ul>\r\n
      \r\n \t
    • 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].<\/li>\r\n \t
    • 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].<\/li>\r\n<\/ul>\r\n \r\n\r\n<\/section>
      For the function below, the local maximum is [latex]16[\/latex], and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].\"Graph<\/section>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.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Definition of a local maximum.[\/caption]\r\n\r\n
      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.[reveal-answer q=\"818075\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"818075\"]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].\"GraphAnalysis of the Solution<\/strong>[latex]\\\\[\/latex]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.\"GraphBased 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.)[\/hidden-answer]<\/section>
      Recall that points on the graph of a function are ordered pairs in the form of\r\n

      [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

      \"the extrema\u00a0occurs at <\/em>[latex]x[\/latex], and that the minimum or maximum\u00a0is\u00a0<\/em>[latex]f(x)[\/latex].\"<\/p>\r\n\r\n<\/section>

      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.[reveal-answer q=\"466198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"466198\"]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].\"Graph\u00a0<\/span>[\/hidden-answer]<\/section>
      [ohm_question hide_question_numbers=1]317887[\/ohm_question]<\/section>\r\n

      Use A Graph to Locate the Absolute Maximum and Absolute Minimum<\/h3>\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 absolute maximum <\/strong>and 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\"Graph\r\n\r\n
      \r\n

      absolute maxima and minima<\/h3>\r\n
        \r\n \t
      • The 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].<\/li>\r\n \t
      • The 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].<\/li>\r\n<\/ul>\r\n<\/section>
        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.<\/section>
        For the function [latex]f[\/latex] shown below, find all absolute maxima and minima.\"Graph\r\n[reveal-answer q=\"461473\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"461473\"]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 y<\/em>-coordinate at [latex]x=-2[\/latex] and [latex]x=2[\/latex], which is [latex]16[\/latex].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 y<\/em>-coordinate at [latex]x=3[\/latex], which is [latex]-10[\/latex].[\/hidden-answer]<\/section>","rendered":"

        Local and Absolute Extrema<\/h2>\n
        \n

        local minima and local maxima (extrema)<\/h3>\n
          \n
        • A function [latex]f[\/latex] is an 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].<\/li>\n
        • A function [latex]f[\/latex] is a 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].<\/li>\n<\/ul>\n
            \n
          • 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].<\/li>\n
          • 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].<\/li>\n<\/ul>\n

             <\/p>\n<\/section>\n

            For the function below, the local maximum is [latex]16[\/latex], and it occurs at [latex]x=-2[\/latex]. The local minimum is [latex]-16[\/latex] and it occurs at [latex]x=2[\/latex].\"Graph<\/section>\n

            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

            \"Graph
            Definition of a local maximum.<\/figcaption><\/figure>\n
            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