{"id":194,"date":"2019-08-27T20:56:15","date_gmt":"2019-08-27T20:56:15","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=194"},"modified":"2020-06-18T16:23:41","modified_gmt":"2020-06-18T16:23:41","slug":"second-derivative","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/second-derivative\/","title":{"rendered":"Second Derivative"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Concavity and the Second Derivative Test &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nYou are learning that calculus is a valuable tool.    One of\nthe most important applications of differential calculus is to find \n<b>extreme function values<\/b>.    The calculus methods for finding the\nmaximum and minimum values of a function are the basic tools\nof <b>optimization theory<\/b>, a very active branch of\nmathematical research   applied    to nearly all\nfields of practical endeavor. Although modern optimization theory\nis considerably more advanced, its methods and fundamental ideas\nclearly show their historical relationship to the calculus. In this\ntutorial you will review how the  second derivative of a\nfunction is related to the shape of its graph and how that information can be\nused to classify relative extreme values.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Some First Derivative Facts<\/h4>\n\n\n\n<p> If you have not already done so, you should review the tutorial on the  <strong><a href=\"https:\/\/math.hmc.edu\/calculus-tutorials\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/first-derivative\/\">First Derivative<\/a><\/strong>. Below is a picture that summarizes the First Derivative Test. <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"512\" height=\"269\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/firstderiv.gif?resize=512%2C269&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-398\"\/><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Concavity<\/h4>\n\n\n\n<p>\n\n  The Second Derivative Test provides a means of classifying\nrelative extreme values by using the sign of the second derivative\nat the critical number. To appreciate this test, it is first\nnecessary to understand the concept of  concavity.\n\n<\/p>\n\n\n\n<p>\n The graph of a function $f$\nis <b>concave upward<\/b> at the point\n$(c,f(c))$ if $f'(c)$ exists and if for all $x$ in some  open interval\ncontaining $c$, the point $(x,f(x))$ on the\n graph of $f$\nlies  above the corresponding point on the graph of the tangent line\nto $f$ at  $c$. This is expressed by the inequality\n $f(x) &gt; [f(c) + f'(c)(x-c)]$ for all\n$x$ in some open interval containing $c$.\n Imagine holding a ruler along the tangent line through the point\n$(c,f(c))$: if the ruler <i>supports the graph of $f$<\/i> near\n$(c,f(c))$, then the graph of the function is concave upward.\n\n<\/p>\n\n\n\n<p>\nThe graph of a function $f$\nis <b>concave downward<\/b> at the point\n$(c,f(c))$ if $f'(c)$ exists and if for all $x$ in some  open interval\ncontaining $c$,  the point $(x,f(x))$ on the\n graph of $f$\nlies  below the corresponding point on the graph of the tangent line\nto $f$ at  $c$. This is expressed by the inequality\n $f(x) &lt; [f(c) + f'(c)(x-c)]$ for all\n$x$ in some open interval containing $c$.\n     In this situation <i>the graph of $f$ supports the ruler<\/i>. This is pictured below:\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"619\" height=\"315\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/rulers.gif?resize=619%2C315&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-399\"\/><\/figure><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Concavity and the Second Derivative<\/h4>\n\n\n\n<p>\n\nThe important result that relates the concavity of the graph of \na function to its\nderivatives is the following one: \n\n<\/p>\n\n\n\n<center>\n<p align=\"center\">\n\t<b>Concavity Theorem<\/b>: If the function \n\t$f$ is twice differentiable at $x=c$, then the graph\n\t of $f$ is concave upward at $(c,f(c))$ if $f&#8221;(c) &gt; 0$ and\n\tconcave downward if $f&#8221;(c) &lt; 0$.\n<\/p>\n<\/center>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nSuppose $f(x) = x^3 -3x^2 + x &#8211; 2.$ Let&#8217;s determine where the\ngraph of $f$ is concave up and where it is concave down. Since $f$ is\ntwice-differentiable for all $x$, we use the result given above and first\ndetermine that $f&#8221;(x) = 6(x -1)$. Thus, $f&#8221;(x) &gt; 0$ if $x &gt; 1$ and $f&#8221;(x) &lt; 0$\nif $x &lt; 1$.  By the Concavity Theorem, the graph of $f$ is concave up for $x &gt;\n1$ and concave down for $x &lt; 1$.       \n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Inflection Points<\/h4>\n\n\n\n<p>\n\nNotice in the example above, that the concavity of the graph of $f$\n<i>changes sign at $x = 1$<\/i>. Points on the graph of $f$ where the\nconcavity changes from up-to-down or down-to-up are called <b>inflection\npoints<\/b> of the graph. The following result connects the concept of inflection\npoints to the derivatives properties of the function:\n\n<\/p>\n\n\n\n<center>\n<p align=\"center\">\n\t<b>Inflection Point Theorem:<\/b> If $f'(c)$ exists and $f&#8221;(c)$\n\t<i>changes sign at $x=c$<\/i>, then the point \n\t$(c,f(c))$ is an <b>inflection point<\/b> of the graph of $f$. If $f&#8221;(c)$ \n\texists at the inflection point, then $f&#8221;(c) = 0$.\n<\/p>\n<\/center>\n\n\n\n<p>\nIf we return to our example, where $f(x) = x^3 &#8211; 3x^2 + x -2$, the <b>\nInflection Point Theorem<\/b> verifies that the graph of $f$ has an inflection\npoint at $x = 1$, since $f&#8221;(1) = 0$. \n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">The Second Derivative Test<\/h4>\n\n\n\n<p>\n\nThe Second Derivative Test relates the concepts of critical points, extreme\nvalues, and concavity to give a very useful tool for determining whether a\ncritical point on the graph of a function is a relative minimum or maximum.\n\n<\/p>\n\n\n\n<center>\n<p align=\"center\">\n\t<b>The Second Derivative Test:<\/b> Suppose that $c$ is a critical point \n\tat which $f'(c)= 0$, that $f'(x)$ exists in a neighborhood of $c$, and\n\tthat $f&#8221;(c)$ exists. Then $f$ has a relative maximum value\n\tat $c$ if $f&#8221;(c) &lt; 0$\n\tand a relative minimum value at $c$ if $f&#8221;(c)&gt; 0$.\n\tIf $f&#8221;(c)= 0$, the test is not informative.\n<\/p>\n<\/center>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nLet&#8217;s find and classify \nthe extreme values for the function \n$f$ with values $f(x) = x^3 -3x^2 + x &#8211; 2$ that was introduced above. We find\nthat $f'(x) = 3x^2 -6x + 1$, and so there are two critical numbers where \n$f'(c) = 0$:\n$$\n     c_1 = 3 &#8211; \\sqrt{6} \\quad {\\rm and} \\quad c_2 = 3 + \\sqrt{6}.\n$$\nNotice that $c_1 &lt; 1$ and that $f&#8221;(c) &lt; 0$. Thus $f$ has a relative maximum at \n$x = 3 &#8211; \\sqrt{6}$. Since $c_2 &gt; 1$ and $f&#8221;(c_2) &gt; 0$, the Second Derivative\nTest informs us that $f$ has a relative minumum at $x = 3 + \\sqrt{6}$.\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> <b>Concavity Theorem<\/b>:\n\t\t<p>\n\t\tIf the function $f$ is twice \n\t\tdifferentiable at $x=c$, then the graph\n\t\t of $f$ is concave upward at $(c,f(c))$ if $f&#8221;(c) &gt; 0$ and\n\t\tconcave downward if $f&#8221;(c) &lt; 0$.\n\t<br><br>\n\t<\/p><\/li><li> <b>The Second Derivative Test<\/b>:\n\t\t<p>\n\t\tSuppose that $c$ is a critical point \n\t\tat which $f'(c)= 0$, that $f'(x)$ exists in a neighborhood of $c$, and\n\t\tthat $f&#8221;(c)$ exists. Then $f$ has a relative maximum value\n\t\tat $c$ if $f&#8221;(c) &lt; 0$\n\t\tand a relative minimum value at $c$ if $f&#8221;(c)&gt; 0$.\n\t\tIf $f&#8221;(c)= 0$, the test is not informative.\n\n<\/p><\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ4510\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Concavity and the Second Derivative Test &#8211; HMC Calculus Tutorial You are learning that calculus is a valuable tool. One of the most important applications of differential calculus is to find extreme function values. The calculus methods for finding the maximum and minimum values of a function are the basic tools of optimization theory, a&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-194","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=194"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/194\/revisions"}],"predecessor-version":[{"id":1220,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/194\/revisions\/1220"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=194"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}