{"id":3268,"date":"2024-06-18T14:36:49","date_gmt":"2024-06-18T14:36:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3268"},"modified":"2024-08-05T02:19:02","modified_gmt":"2024-08-05T02:19:02","slug":"derivatives-and-the-shape-of-a-graph-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-and-the-shape-of-a-graph-fresh-take\/","title":{"raw":"Derivatives and the Shape of a Graph: Fresh Take","rendered":"Derivatives and the Shape of a Graph: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Use the first derivative to determine where a function is going up or down, and identify points that might be local highs or lows<\/li>\r\n\t<li>Apply the second derivative to find out where a function curves upward or downward and locate points where this curvature changes<\/li>\r\n\t<li>Use the second derivative test to determine if a point on a graph is the highest or lowest within specific sections<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>The First Derivative Test<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Used to determine local extrema of a continuous function<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Theoretical Basis:\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 function changes from increasing to decreasing (or vice versa) at local extrema<\/li>\r\n\t<li class=\"whitespace-normal break-words\">This change occurs at critical 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\">Test Conditions:\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 continuous over an interval containing the critical point [latex]c[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is differentiable over the interval, except possibly at [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Results:\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: [latex]f'(x)[\/latex] changes from positive to negative at [latex]c[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Local minimum: [latex]f'(x)[\/latex] changes from negative to positive at [latex]c[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Neither: [latex]f'(x)[\/latex] doesn't change sign at [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-bold\"><strong>How to Apply the First Derivative Test<\/strong><\/p>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Find critical points: Solve [latex]f'(x) = 0[\/latex] and where [latex]f'(x)[\/latex] is undefined<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Divide the domain into intervals using the critical points<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Determine the sign of [latex]f'(x)[\/latex] in each interval<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Analyze sign changes to identify local extrema<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>Use the first derivative test to locate all local extrema for [latex]f(x)=\u2212x^3+\\frac{3}{2}x^2+18x[\/latex].<\/p>\r\n<p>[reveal-answer q=\"6192003\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"6192003\"]<\/p>\r\n<p>Find all critical points of [latex]f[\/latex] and determine the signs of [latex]f^{\\prime}(x)[\/latex] over particular intervals determined by the critical points.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043281485\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043281485\"]<\/p>\r\n<p>[latex]f[\/latex] has a local minimum at [latex]-2[\/latex] and a local maximum at [latex]3[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Use the first derivative test to find all local extrema for [latex]f(x)=\\sqrt[3]{x-1}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"25579900\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"25579900\"]<\/p>\r\n<p>The only critical point of [latex]f[\/latex] is [latex]x=1[\/latex].<br \/>\r\n[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042982077\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042982077\"]<\/p>\r\n<p>[latex]f[\/latex] has no local extrema because [latex]f^{\\prime}[\/latex] does not change sign at [latex]x=1[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Concavity and Points of Inflection<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Concavity:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Concave Up: [latex]f'(x)[\/latex] is increasing; [latex]f''(x) &gt; 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Concave Down: [latex]f'(x)[\/latex] is decreasing; [latex]f''(x) &lt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Concavity Test: For a twice-differentiable function [latex]f(x)[\/latex] on interval [latex]I[\/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\">If [latex]f''(x) &gt; 0[\/latex] for all [latex]x \\in I[\/latex], [latex]f[\/latex] is concave up on [latex]I[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">If [latex]f''(x) &lt; 0[\/latex] for all [latex]x \\in I[\/latex], [latex]f[\/latex] is concave down on [latex]I[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Inflection 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 concavity changes<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Occur where [latex]f''(x) = 0[\/latex] or [latex]f''(x)[\/latex] is undefined<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] must be continuous at these points<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Relationship to Derivatives:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">First derivative ([latex]f'[\/latex]) indicates increasing\/decreasing<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Second derivative ([latex]f''[\/latex]) indicates concavity<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-bold\"><strong>How to Analyze Concavity and Find Inflection Points<\/strong><\/p>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Find [latex]f''(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Determine where [latex]f''(x) = 0[\/latex] or is undefined<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use these points to divide the domain into intervals<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Test the sign of [latex]f''(x)[\/latex] in each interval<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Identify where concavity changes to locate inflection points<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p>For [latex]f(x)=\u2212x^3+\\frac{3}{2}x^2+18x[\/latex], find all intervals where [latex]f[\/latex] is concave up and all intervals where [latex]f[\/latex] is concave down.<\/p>\r\n<p>[reveal-answer q=\"33881102\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"33881102\"]<\/p>\r\n<p>Find where [latex]f^{\\prime \\prime}(x)=0[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042369709\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042369709\"]<\/p>\r\n<p>[latex]f[\/latex] is concave up over the interval [latex](\u2212\\infty ,\\frac{1}{2})[\/latex] and concave down over the interval [latex](\\frac{1}{2},\\infty )[\/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\">Analyze the concavity and find inflection points of [latex]f(x) = x^3 - 3x^2 + 1[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\"><br \/>\r\n[reveal-answer q=\"269091\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"269091\"]<\/p>\r\n<p>Find [latex]f''(x)[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'(x) = 3x^2 - 6x[\/latex]<br \/>\r\n[latex]f''(x) = 6x - 6 = 6(x - 1)[\/latex]<\/p>\r\n<p>Solve [latex]f''(x) = 0[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]6(x - 1) = 0[\/latex]<br \/>\r\n[latex]x = 1[\/latex]<\/p>\r\n<p>Create sign chart for [latex]f''(x)[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{|c|c|c|}<br \/>\r\n\\hline<br \/>\r\n\\text{Interval} &amp; (-\\infty, 1) &amp; (1, \\infty) \\\\<br \/>\r\n\\hline<br \/>\r\n\\text{Sign of } f''(x) &amp; - &amp; + \\\\<br \/>\r\n\\hline<br \/>\r\n\\text{Concavity} &amp; \\text{Concave down} &amp; \\text{Concave up} \\\\<br \/>\r\n\\hline<br \/>\r\n\\end{array}[\/latex]<\/p>\r\n<p>Analyze:<\/p>\r\n<p>Concavity changes at [latex]x = 1[\/latex]<br \/>\r\n[latex]f(1) = 1 - 3 + 1 = -1[\/latex]<\/p>\r\n<p>Conclusion:<\/p>\r\n<p>Concave down on [latex](-\\infty, 1)[\/latex]<br \/>\r\nConcave up on [latex](1, \\infty)[\/latex]<br \/>\r\nInflection point at [latex](1, -1)[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>The Second Derivative Test<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Purpose:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Used to determine the nature of critical points (local maxima or minima)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Theorem Statement: For a twice-differentiable function [latex]f(x)[\/latex] with [latex]f'(c) = 0[\/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\">If [latex]f''(c) &gt; 0[\/latex], [latex]f[\/latex] has a local minimum at [latex]c[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">If [latex]f''(c) &lt; 0[\/latex], [latex]f[\/latex] has a local maximum at [latex]c[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">If [latex]f''(c) = 0[\/latex], the test is inconclusive<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Relationship to Concavity:\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) &gt; 0[\/latex] indicates concave up at [latex]c[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f''(c) &lt; 0[\/latex] indicates concave down at [latex]c[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Advantages:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Often simpler than the First Derivative Test<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Provides information about concavity<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Limitations:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Requires [latex]f''(x)[\/latex] to be continuous near [latex]c[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Inconclusive when [latex]f''(c) = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"font-bold\"><strong>How to Apply the Second Derivative Test<\/strong><\/p>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Find critical points by solving [latex]f'(x) = 0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Calculate [latex]f''(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Evaluate [latex]f''(x)[\/latex] at each critical point<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the test to classify each critical point<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For inconclusive cases, use the First Derivative Test<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">\r\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>\r\n<p>[reveal-answer q=\"707252\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"707252\"]<\/p>\r\n<p>[latex]f^{\\prime \\prime}(x)=6x-3[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043173990\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043173990\"]<\/p>\r\n<p>[latex]f[\/latex] has a local maximum at -2 and a local minimum at 3.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find and classify the local extrema of [latex]f(x) = x^4 - 4x^3 + 2[\/latex] using the Second Derivative Test.<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"959759\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"959759\"]<\/p>\r\n<p>Find [latex]f'(x)[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3)[\/latex]<\/p>\r\n<p>Find critical points:<\/p>\r\n<p style=\"text-align: center;\">[latex]f'(x) = 0[\/latex] when [latex]x = 0[\/latex] or [latex]x = 3[\/latex]<\/p>\r\n<p>Calculate [latex]f''(x)[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]f''(x) = 12x^2 - 24x = 12x(x - 2)[\/latex]<\/p>\r\n<p>Evaluate [latex]f''(x)[\/latex] at critical points:<\/p>\r\n<p>At [latex]x = 0[\/latex]: [latex]f''(0) = 0[\/latex] (inconclusive)<br \/>\r\nAt [latex]x = 3[\/latex]: [latex]f''(3) = 12(3)(1) = 36 &gt; 0[\/latex] (local minimum)<\/p>\r\n<p>For [latex]x = 0[\/latex], use First Derivative Test:<\/p>\r\n<p>[latex]f'(x) = 4x^2(x - 3)[\/latex] is positive for [latex]x[\/latex] near [latex]0[\/latex], so neither max nor min<\/p>\r\n<p>Conclusion:<\/p>\r\n<p>Local minimum at [latex]x = 3[\/latex]<br \/>\r\nNo local extremum at [latex]x = 0[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use the first derivative to determine where a function is going up or down, and identify points that might be local highs or lows<\/li>\n<li>Apply the second derivative to find out where a function curves upward or downward and locate points where this curvature changes<\/li>\n<li>Use the second derivative test to determine if a point on a graph is the highest or lowest within specific sections<\/li>\n<\/ul>\n<\/section>\n<h2>The First Derivative Test<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used to determine local extrema of a continuous function<\/li>\n<li class=\"whitespace-normal break-words\">Theoretical Basis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A function changes from increasing to decreasing (or vice versa) at local extrema<\/li>\n<li class=\"whitespace-normal break-words\">This change occurs at critical 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\">Test Conditions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is continuous over an interval containing the critical point [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] is differentiable over the interval, except possibly at [latex]c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Results:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Local maximum: [latex]f'(x)[\/latex] changes from positive to negative at [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Local minimum: [latex]f'(x)[\/latex] changes from negative to positive at [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Neither: [latex]f'(x)[\/latex] doesn&#8217;t change sign at [latex]c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>How to Apply the First Derivative Test<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find critical points: Solve [latex]f'(x) = 0[\/latex] and where [latex]f'(x)[\/latex] is undefined<\/li>\n<li class=\"whitespace-normal break-words\">Divide the domain into intervals using the critical points<\/li>\n<li class=\"whitespace-normal break-words\">Determine the sign of [latex]f'(x)[\/latex] in each interval<\/li>\n<li class=\"whitespace-normal break-words\">Analyze sign changes to identify local extrema<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p>Use the first derivative test to locate all local extrema for [latex]f(x)=\u2212x^3+\\frac{3}{2}x^2+18x[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q6192003\">Hint<\/button><\/p>\n<div id=\"q6192003\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find all critical points of [latex]f[\/latex] and determine the signs of [latex]f^{\\prime}(x)[\/latex] over particular intervals determined by the critical points.<\/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-id1165043281485\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043281485\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f[\/latex] has a local minimum at [latex]-2[\/latex] and a local maximum at [latex]3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Use the first derivative test to find all local extrema for [latex]f(x)=\\sqrt[3]{x-1}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q25579900\">Hint<\/button><\/p>\n<div id=\"q25579900\" class=\"hidden-answer\" style=\"display: none\">\n<p>The only critical point of [latex]f[\/latex] is [latex]x=1[\/latex].\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042982077\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042982077\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f[\/latex] has no local extrema because [latex]f^{\\prime}[\/latex] does not change sign at [latex]x=1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Concavity and Points of Inflection<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Concavity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Concave Up: [latex]f'(x)[\/latex] is increasing; [latex]f''(x) > 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Concave Down: [latex]f'(x)[\/latex] is decreasing; [latex]f''(x) < 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Concavity Test: For a twice-differentiable function [latex]f(x)[\/latex] on interval [latex]I[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]f''(x) > 0[\/latex] for all [latex]x \\in I[\/latex], [latex]f[\/latex] is concave up on [latex]I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]f''(x) < 0[\/latex] for all [latex]x \\in I[\/latex], [latex]f[\/latex] is concave down on [latex]I[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Inflection Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Points where concavity changes<\/li>\n<li class=\"whitespace-normal break-words\">Occur where [latex]f''(x) = 0[\/latex] or [latex]f''(x)[\/latex] is undefined<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(x)[\/latex] must be continuous at these points<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship to Derivatives:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">First derivative ([latex]f'[\/latex]) indicates increasing\/decreasing<\/li>\n<li class=\"whitespace-normal break-words\">Second derivative ([latex]f''[\/latex]) indicates concavity<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>How to Analyze Concavity and Find Inflection Points<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find [latex]f''(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Determine where [latex]f''(x) = 0[\/latex] or is undefined<\/li>\n<li class=\"whitespace-normal break-words\">Use these points to divide the domain into intervals<\/li>\n<li class=\"whitespace-normal break-words\">Test the sign of [latex]f''(x)[\/latex] in each interval<\/li>\n<li class=\"whitespace-normal break-words\">Identify where concavity changes to locate inflection points<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p>For [latex]f(x)=\u2212x^3+\\frac{3}{2}x^2+18x[\/latex], find all intervals where [latex]f[\/latex] is concave up and all intervals where [latex]f[\/latex] is concave down.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q33881102\">Hint<\/button><\/p>\n<div id=\"q33881102\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find where [latex]f^{\\prime \\prime}(x)=0[\/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-id1165042369709\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042369709\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f[\/latex] is concave up over the interval [latex](\u2212\\infty ,\\frac{1}{2})[\/latex] and concave down over the interval [latex](\\frac{1}{2},\\infty )[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Analyze the concavity and find inflection points of [latex]f(x) = x^3 - 3x^2 + 1[\/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=\"q269091\">Show Answer<\/button><\/p>\n<div id=\"q269091\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find [latex]f''(x)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]f'(x) = 3x^2 - 6x[\/latex]<br \/>\n[latex]f''(x) = 6x - 6 = 6(x - 1)[\/latex]<\/p>\n<p>Solve [latex]f''(x) = 0[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]6(x - 1) = 0[\/latex]<br \/>\n[latex]x = 1[\/latex]<\/p>\n<p>Create sign chart for [latex]f''(x)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{|c|c|c|}<br \/>  \\hline<br \/>  \\text{Interval} & (-\\infty, 1) & (1, \\infty) \\\\<br \/>  \\hline<br \/>  \\text{Sign of } f''(x) & - & + \\\\<br \/>  \\hline<br \/>  \\text{Concavity} & \\text{Concave down} & \\text{Concave up} \\\\<br \/>  \\hline<br \/>  \\end{array}[\/latex]<\/p>\n<p>Analyze:<\/p>\n<p>Concavity changes at [latex]x = 1[\/latex]<br \/>\n[latex]f(1) = 1 - 3 + 1 = -1[\/latex]<\/p>\n<p>Conclusion:<\/p>\n<p>Concave down on [latex](-\\infty, 1)[\/latex]<br \/>\nConcave up on [latex](1, \\infty)[\/latex]<br \/>\nInflection point at [latex](1, -1)[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\"><\/div>\n<\/div>\n<\/section>\n<h2>The Second Derivative Test<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Purpose:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used to determine the nature of critical points (local maxima or minima)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Theorem Statement: For a twice-differentiable function [latex]f(x)[\/latex] with [latex]f'(c) = 0[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]f''(c) > 0[\/latex], [latex]f[\/latex] has a local minimum at [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]f''(c) < 0[\/latex], [latex]f[\/latex] has a local maximum at [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]f''(c) = 0[\/latex], the test is inconclusive<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship to Concavity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]f''(c) > 0[\/latex] indicates concave up at [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f''(c) < 0[\/latex] indicates concave down at [latex]c[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Advantages:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Often simpler than the First Derivative Test<\/li>\n<li class=\"whitespace-normal break-words\">Provides information about concavity<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Limitations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Requires [latex]f''(x)[\/latex] to be continuous near [latex]c[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Inconclusive when [latex]f''(c) = 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"font-bold\"><strong>How to Apply the Second Derivative Test<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find critical points by solving [latex]f'(x) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]f''(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Evaluate [latex]f''(x)[\/latex] at each critical point<\/li>\n<li class=\"whitespace-normal break-words\">Use the test to classify each critical point<\/li>\n<li class=\"whitespace-normal break-words\">For inconclusive cases, use the First Derivative Test<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q707252\">Hint<\/button><\/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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043173990\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find and classify the local extrema of [latex]f(x) = x^4 - 4x^3 + 2[\/latex] using the Second Derivative Test.<\/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=\"q959759\">Show Answer<\/button><\/p>\n<div id=\"q959759\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find [latex]f'(x)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3)[\/latex]<\/p>\n<p>Find critical points:<\/p>\n<p style=\"text-align: center;\">[latex]f'(x) = 0[\/latex] when [latex]x = 0[\/latex] or [latex]x = 3[\/latex]<\/p>\n<p>Calculate [latex]f''(x)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]f''(x) = 12x^2 - 24x = 12x(x - 2)[\/latex]<\/p>\n<p>Evaluate [latex]f''(x)[\/latex] at critical points:<\/p>\n<p>At [latex]x = 0[\/latex]: [latex]f''(0) = 0[\/latex] (inconclusive)<br \/>\nAt [latex]x = 3[\/latex]: [latex]f''(3) = 12(3)(1) = 36 > 0[\/latex] (local minimum)<\/p>\n<p>For [latex]x = 0[\/latex], use First Derivative Test:<\/p>\n<p>[latex]f'(x) = 4x^2(x - 3)[\/latex] is positive for [latex]x[\/latex] near [latex]0[\/latex], so neither max nor min<\/p>\n<p>Conclusion:<\/p>\n<p>Local minimum at [latex]x = 3[\/latex]<br \/>\nNo local extremum at [latex]x = 0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":29,"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\/3268"}],"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\/3268\/revisions"}],"predecessor-version":[{"id":3836,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3268\/revisions\/3836"}],"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\/3268\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3268"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3268"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3268"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3268"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}