{"id":3233,"date":"2024-06-13T18:55:27","date_gmt":"2024-06-13T18:55:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3233"},"modified":"2024-08-05T02:15:40","modified_gmt":"2024-08-05T02:15:40","slug":"maxima-and-minima-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/maxima-and-minima-fresh-take\/","title":{"raw":"Maxima and Minima: Fresh Take","rendered":"Maxima and Minima: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Define and Identify the highest and lowest points of a function on a graph, both overall and within specific sections<\/li>\r\n\t<li>Locate points on a function within a specific range where the slope is zero or undefined (critical points)<\/li>\r\n\t<li>Explain how to use critical points to find the highest or lowest values of a function within a limited range<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Extrema and Critical Points<\/h2>\r\n<h3>Absolute Extrema<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Absolute Extrema:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Absolute Maximum:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(c) \\ge f(x)[\/latex] for all [latex]x[\/latex] in the interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Absolute Minimum:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(c) \\le f(x)[\/latex] for all [latex]x[\/latex] in the interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Can be positive, negative, or zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Extreme Value Theorem:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For a continuous function [latex]f[\/latex] on a closed interval [latex][a,b][\/latex]:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">There exists a point where [latex]f[\/latex] has an absolute maximum<\/li>\r\n\t<li class=\"whitespace-normal break-words\">There exists a point where [latex]f[\/latex] has an absolute minimum<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Conditions for Extrema:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Continuity is crucial for the existence of extrema<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Closed, bounded intervals guarantee extrema for continuous functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Types of Domains:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Infinite intervals may lack extrema<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Open intervals may lack extrema even for continuous functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Discontinuities can prevent extrema from existing<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Analyze the function [latex]f(x) = x^3 - 3x[\/latex] on the interval [latex][-2, 2][\/latex] for absolute extrema.<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"554901\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"554901\"]<\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Check if the Extreme Value Theorem applies:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is a polynomial, so it's continuous<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The interval [latex][-2, 2][\/latex] is closed and bounded<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Therefore, absolute extrema must exist on this interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Find critical points in [latex][-2, 2][\/latex]:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f'(x) = 3x^2 - 3 = 3(x^2 - 1)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Set [latex]f'(x) = 0[\/latex]: [latex]x^2 = 1[\/latex], so [latex]x = \\pm 1[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Both critical points are in the interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Evaluate [latex]f(x)[\/latex] at critical points and endpoints:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(-2) = -8 + 6 = -2[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(-1) = -1 + 3 = 2[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(0) = 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(1) = 1 - 3 = -2[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(2) = 8 - 6 = 2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Identify extrema:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Absolute maximum: [latex]2[\/latex], occurs at [latex]x = -1[\/latex] and [latex]x = 2[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Absolute minimum: [latex]-2[\/latex], occurs at [latex]x = -2[\/latex] and [latex]x = 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3>Local Extrema and Critical Points<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Local Extrema:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Local Maximum:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(c) \\ge f(x)[\/latex] for all [latex]x[\/latex] in some open interval containing [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Local Minimum:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(c) \\le f(x)[\/latex] for all [latex]x[\/latex] in some open interval containing [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Critical Points:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Interior point [latex]c[\/latex] where [latex]f'(c) = 0[\/latex] or [latex]f'(c)[\/latex] is undefined<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Candidates for local extrema<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Fermat's Theorem:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">If [latex]f[\/latex] has a local extremum at [latex]c[\/latex] and [latex]f[\/latex] is differentiable at [latex]c[\/latex], then [latex]f'(c) = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Relationships:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Local extrema occur only at critical points<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Not all critical points yield local extrema<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Endpoint extrema are not considered local extrema<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Types of Critical Points:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Horizontal tangent line: [latex]f'(c) = 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical tangent line or corner point: [latex]f'(c)[\/latex] undefined<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Find all critical points for [latex]f(x)=x^3-\\frac{1}{2}x^2-2x+1[\/latex].<\/p>\r\n<p>[reveal-answer q=\"30077654\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"30077654\"]<\/p>\r\n<p id=\"fs-id1165040757635\">Calculate [latex]f^{\\prime}(x)[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165040757600\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165040757600\"]<\/p>\r\n<p>[latex]x=-\\frac{2}{3}[\/latex], [latex]x=1[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find the critical points of [latex]f(x) = x^{2\/3}(x-2)[\/latex] and determine if they correspond to local extrema.<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"61328\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"61328\"]<\/p>\r\n<p>Find [latex]f'(x)[\/latex] using the product rule:<\/p>\r\n<p>[latex]f'(x) = \\frac{2}{3}x^{-1\/3}(x-2) + x^{2\/3}[\/latex]<br \/>\r\n[latex]f'(x) = \\frac{2}{3}x^{-1\/3}(x-2) + x^{2\/3} \\cdot 1[\/latex]<\/p>\r\n<p>Simplify:<\/p>\r\n<p>[latex]f'(x) = \\frac{2x^{2\/3} - 4x^{-1\/3} + 3x^{2\/3}}{3x^{1\/3}}[\/latex]<br \/>\r\n[latex]f'(x) = \\frac{5x^{2\/3} - 4x^{-1\/3}}{3x^{1\/3}}[\/latex]<\/p>\r\n<p>Find critical points:<\/p>\r\n<p>a) Where [latex]f'(x) = 0[\/latex]:<\/p>\r\n<p>[latex]5x^{2\/3} - 4x^{-1\/3} = 0[\/latex]<br \/>\r\n[latex]5x - 4 = 0[\/latex]<br \/>\r\n[latex]x = \\frac{4}{5}[\/latex]<\/p>\r\n<p>b) Where [latex]f'(x)[\/latex] is undefined:<\/p>\r\n<p>[latex]x = 0[\/latex] (due to division by [latex]x^{1\/3}[\/latex])<\/p>\r\n<p>Evaluate behavior at critical points:<\/p>\r\n<p>At [latex]x = 0[\/latex]: function has a vertical tangent line (cusp)<br \/>\r\nAt [latex]x = \\frac{4}{5}[\/latex]: function has a horizontal tangent line<\/p>\r\n<p>Conclusion:<\/p>\r\n<p>[latex]x = 0[\/latex] is a critical point but not a local extremum (inflection point)<br \/>\r\n[latex]x = \\frac{4}{5}[\/latex] is a critical point and a local minimum<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h3>Locating Absolute Extrema<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Extreme Value Theorem:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">A continuous function on a closed, bounded interval has both an absolute maximum and minimum<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Location of Absolute Extrema:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Occur at endpoints of the interval or at critical points within the interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Critical Points:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Points where [latex]f'(x) = 0[\/latex] or [latex]f'(x)[\/latex] is undefined<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Procedure for Finding Absolute Extrema:\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Evaluate function at endpoints\u00a0<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Find and evaluate function at critical points within the interval\u00a0<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compare all values to determine absolute maximum and minimum<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Interior vs. Endpoint Extrema:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Absolute extrema at interior points are also local extrema<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Endpoint extrema are not considered local extrema<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Find the absolute maximum and absolute minimum of [latex]f(x)=x^2-4x+3[\/latex] over the interval [latex][1,4][\/latex].<\/p>\r\n<p>[reveal-answer q=\"4227188\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"4227188\"]<\/p>\r\n<p>Look for critical points. Evaluate [latex]f[\/latex] at all critical points and at the endpoints.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042108934\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042108934\"]<\/p>\r\n<p>The absolute maximum is 3 and it occurs at [latex]x=4[\/latex]. The absolute minimum is -1 and it occurs at [latex]x=2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the absolute extrema of [latex]f(x) = x^3 - 6x^2 + 9x[\/latex] on the interval [latex][0,4][\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"495071\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"495071\"]<\/p>\r\n<p>Evaluate at endpoints:<\/p>\r\n<p>[latex]f(0) = 0[\/latex]<br \/>\r\n[latex]f(4) = 64 - 96 + 36 = 4[\/latex]<\/p>\r\n<p>Find critical points:<\/p>\r\n<p>[latex]f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x - 1)(x - 3)[\/latex]<\/p>\r\n<p>Critical points:<\/p>\r\n<p>[latex]x = 1[\/latex] and [latex]x = 3[\/latex] (both within [latex][0,4][\/latex])<\/p>\r\n<p>Evaluate at critical points:<\/p>\r\n<p>[latex]f(1) = 1 - 6 + 9 = 4[\/latex]<br \/>\r\n[latex]f(3) = 27 - 54 + 27 = 0[\/latex]<\/p>\r\n<p>Compare all values:<\/p>\r\n<p>[latex]\\begin{array}{|c|c|}<br \/>\r\n\\hline<br \/>\r\nx &amp; f(x) \\\\<br \/>\r\n\\hline<br \/>\r\n0 &amp; 0 \\\\<br \/>\r\n1 &amp; 4 \\\\<br \/>\r\n3 &amp; 0 \\\\<br \/>\r\n4 &amp; 4 \\\\<br \/>\r\n\\hline<br \/>\r\n\\end{array}[\/latex]<\/p>\r\n<p>Conclusion:<\/p>\r\n<p>Absolute maximum: [latex]4[\/latex], occurs at [latex]x = 1[\/latex] and [latex]x = 4[\/latex]<br \/>\r\nAbsolute minimum: [latex]0[\/latex], occurs at [latex]x = 0[\/latex] and [latex]x = 3[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Define and Identify the highest and lowest points of a function on a graph, both overall and within specific sections<\/li>\n<li>Locate points on a function within a specific range where the slope is zero or undefined (critical points)<\/li>\n<li>Explain how to use critical points to find the highest or lowest values of a function within a limited range<\/li>\n<\/ul>\n<\/section>\n<h2>Extrema and Critical Points<\/h2>\n<h3>Absolute Extrema<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Absolute Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Absolute Maximum:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(c) \\ge f(x)[\/latex] for all [latex]x[\/latex] in the interval<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Absolute Minimum:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(c) \\le f(x)[\/latex] for all [latex]x[\/latex] in the interval<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Can be positive, negative, or zero<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Extreme Value Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For a continuous function [latex]f[\/latex] on a closed interval [latex][a,b][\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">There exists a point where [latex]f[\/latex] has an absolute maximum<\/li>\n<li class=\"whitespace-normal break-words\">There exists a point where [latex]f[\/latex] has an absolute minimum<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conditions for Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Continuity is crucial for the existence of extrema<\/li>\n<li class=\"whitespace-normal break-words\">Closed, bounded intervals guarantee extrema for continuous functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Domains:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Infinite intervals may lack extrema<\/li>\n<li class=\"whitespace-normal break-words\">Open intervals may lack extrema even for continuous functions<\/li>\n<li class=\"whitespace-normal break-words\">Discontinuities can prevent extrema from existing<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Analyze the function [latex]f(x) = x^3 - 3x[\/latex] on the interval [latex][-2, 2][\/latex] for absolute extrema.<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q554901\">Show Answer<\/button><\/p>\n<div id=\"q554901\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Check if the Extreme Value Theorem applies:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is a polynomial, so it&#8217;s continuous<\/li>\n<li class=\"whitespace-normal break-words\">The interval [latex][-2, 2][\/latex] is closed and bounded<\/li>\n<li class=\"whitespace-normal break-words\">Therefore, absolute extrema must exist on this interval<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Find critical points in [latex][-2, 2][\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f'(x) = 3x^2 - 3 = 3(x^2 - 1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set [latex]f'(x) = 0[\/latex]: [latex]x^2 = 1[\/latex], so [latex]x = \\pm 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Both critical points are in the interval<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluate [latex]f(x)[\/latex] at critical points and endpoints:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(-2) = -8 + 6 = -2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(-1) = -1 + 3 = 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(0) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(1) = 1 - 3 = -2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(2) = 8 - 6 = 2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Identify extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Absolute maximum: [latex]2[\/latex], occurs at [latex]x = -1[\/latex] and [latex]x = 2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Absolute minimum: [latex]-2[\/latex], occurs at [latex]x = -2[\/latex] and [latex]x = 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<h3>Local Extrema and Critical Points<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Local Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Local Maximum:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(c) \\ge f(x)[\/latex] for all [latex]x[\/latex] in some open interval containing [latex]c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Local Minimum:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(c) \\le f(x)[\/latex] for all [latex]x[\/latex] in some open interval containing [latex]c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Critical Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Interior point [latex]c[\/latex] where [latex]f'(c) = 0[\/latex] or [latex]f'(c)[\/latex] is undefined<\/li>\n<li class=\"whitespace-normal break-words\">Candidates for local extrema<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Fermat&#8217;s Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]f[\/latex] has a local extremum at [latex]c[\/latex] and [latex]f[\/latex] is differentiable at [latex]c[\/latex], then [latex]f'(c) = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Relationships:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Local extrema occur only at critical points<\/li>\n<li class=\"whitespace-normal break-words\">Not all critical points yield local extrema<\/li>\n<li class=\"whitespace-normal break-words\">Endpoint extrema are not considered local extrema<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Critical Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Horizontal tangent line: [latex]f'(c) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical tangent line or corner point: [latex]f'(c)[\/latex] undefined<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find all critical points for [latex]f(x)=x^3-\\frac{1}{2}x^2-2x+1[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q30077654\">Hint<\/button><\/p>\n<div id=\"q30077654\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165040757635\">Calculate [latex]f^{\\prime}(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165040757600\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165040757600\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=-\\frac{2}{3}[\/latex], [latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the critical points of [latex]f(x) = x^{2\/3}(x-2)[\/latex] and determine if they correspond to local extrema.<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q61328\">Show Answer<\/button><\/p>\n<div id=\"q61328\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find [latex]f'(x)[\/latex] using the product rule:<\/p>\n<p>[latex]f'(x) = \\frac{2}{3}x^{-1\/3}(x-2) + x^{2\/3}[\/latex]<br \/>\n[latex]f'(x) = \\frac{2}{3}x^{-1\/3}(x-2) + x^{2\/3} \\cdot 1[\/latex]<\/p>\n<p>Simplify:<\/p>\n<p>[latex]f'(x) = \\frac{2x^{2\/3} - 4x^{-1\/3} + 3x^{2\/3}}{3x^{1\/3}}[\/latex]<br \/>\n[latex]f'(x) = \\frac{5x^{2\/3} - 4x^{-1\/3}}{3x^{1\/3}}[\/latex]<\/p>\n<p>Find critical points:<\/p>\n<p>a) Where [latex]f'(x) = 0[\/latex]:<\/p>\n<p>[latex]5x^{2\/3} - 4x^{-1\/3} = 0[\/latex]<br \/>\n[latex]5x - 4 = 0[\/latex]<br \/>\n[latex]x = \\frac{4}{5}[\/latex]<\/p>\n<p>b) Where [latex]f'(x)[\/latex] is undefined:<\/p>\n<p>[latex]x = 0[\/latex] (due to division by [latex]x^{1\/3}[\/latex])<\/p>\n<p>Evaluate behavior at critical points:<\/p>\n<p>At [latex]x = 0[\/latex]: function has a vertical tangent line (cusp)<br \/>\nAt [latex]x = \\frac{4}{5}[\/latex]: function has a horizontal tangent line<\/p>\n<p>Conclusion:<\/p>\n<p>[latex]x = 0[\/latex] is a critical point but not a local extremum (inflection point)<br \/>\n[latex]x = \\frac{4}{5}[\/latex] is a critical point and a local minimum<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3>Locating Absolute Extrema<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Extreme Value Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A continuous function on a closed, bounded interval has both an absolute maximum and minimum<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Location of Absolute Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Occur at endpoints of the interval or at critical points within the interval<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Critical Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Points where [latex]f'(x) = 0[\/latex] or [latex]f'(x)[\/latex] is undefined<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Procedure for Finding Absolute Extrema:\n<ul>\n<li class=\"whitespace-normal break-words\">Evaluate function at endpoints\u00a0<\/li>\n<li class=\"whitespace-normal break-words\">Find and evaluate function at critical points within the interval\u00a0<\/li>\n<li class=\"whitespace-normal break-words\">Compare all values to determine absolute maximum and minimum<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interior vs. Endpoint Extrema:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Absolute extrema at interior points are also local extrema<\/li>\n<li class=\"whitespace-normal break-words\">Endpoint extrema are not considered local extrema<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p>Find the absolute maximum and absolute minimum of [latex]f(x)=x^2-4x+3[\/latex] over the interval [latex][1,4][\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q4227188\">Hint<\/button><\/p>\n<div id=\"q4227188\" class=\"hidden-answer\" style=\"display: none\">\n<p>Look for critical points. Evaluate [latex]f[\/latex] at all critical points and at the endpoints.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042108934\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042108934\" class=\"hidden-answer\" style=\"display: none\">\n<p>The absolute maximum is 3 and it occurs at [latex]x=4[\/latex]. The absolute minimum is -1 and it occurs at [latex]x=2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the absolute extrema of [latex]f(x) = x^3 - 6x^2 + 9x[\/latex] on the interval [latex][0,4][\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q495071\">Show Answer<\/button><\/p>\n<div id=\"q495071\" class=\"hidden-answer\" style=\"display: none\">\n<p>Evaluate at endpoints:<\/p>\n<p>[latex]f(0) = 0[\/latex]<br \/>\n[latex]f(4) = 64 - 96 + 36 = 4[\/latex]<\/p>\n<p>Find critical points:<\/p>\n<p>[latex]f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x - 1)(x - 3)[\/latex]<\/p>\n<p>Critical points:<\/p>\n<p>[latex]x = 1[\/latex] and [latex]x = 3[\/latex] (both within [latex][0,4][\/latex])<\/p>\n<p>Evaluate at critical points:<\/p>\n<p>[latex]f(1) = 1 - 6 + 9 = 4[\/latex]<br \/>\n[latex]f(3) = 27 - 54 + 27 = 0[\/latex]<\/p>\n<p>Compare all values:<\/p>\n<p>[latex]\\begin{array}{|c|c|}<br \/>  \\hline<br \/>  x & f(x) \\\\<br \/>  \\hline<br \/>  0 & 0 \\\\<br \/>  1 & 4 \\\\<br \/>  3 & 0 \\\\<br \/>  4 & 4 \\\\<br \/>  \\hline<br \/>  \\end{array}[\/latex]<\/p>\n<p>Conclusion:<\/p>\n<p>Absolute maximum: [latex]4[\/latex], occurs at [latex]x = 1[\/latex] and [latex]x = 4[\/latex]<br \/>\nAbsolute minimum: [latex]0[\/latex], occurs at [latex]x = 0[\/latex] and [latex]x = 3[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":652,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3233"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3233\/revisions"}],"predecessor-version":[{"id":3833,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3233\/revisions\/3833"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/652"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3233\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3233"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3233"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3233"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}