{"id":309,"date":"2023-09-20T22:48:59","date_gmt":"2023-09-20T22:48:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/the-second-derivative-test\/"},"modified":"2024-08-05T13:13:11","modified_gmt":"2024-08-05T13:13:11","slug":"derivatives-and-the-shape-of-a-graph-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-and-the-shape-of-a-graph-learn-it-3\/","title":{"raw":"Derivatives and the Shape of a Graph: Learn It 3","rendered":"Derivatives and the Shape of a Graph: Learn It 3"},"content":{"raw":"<h2>The Second Derivative Test<\/h2>\r\n<p>The first derivative test provides a systematic approach to identify local extrema, but in some cases, using the second derivative can be more straightforward. A function must have a local extremum at a critical point, but not all critical points are extremas.<\/p>\r\n<p>Consider a function [latex]f[\/latex] that is twice-differentiable on an open interval [latex]I[\/latex] containing [latex]a[\/latex].<\/p>\r\n<ul>\r\n\t<li>If [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] and [latex]f^{\\prime}(a)=0[\/latex], [latex]f[\/latex]is concave down at [latex]a[\/latex], indicating a local maximum.<\/li>\r\n\t<li>If [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] and [latex]f^{\\prime}(a)=0[\/latex], [latex]f[\/latex]is concave up at [latex]a[\/latex], suggesting a local minimum at [latex]a[\/latex].<\/li>\r\n<\/ul>\r\n<p>Furthermore, if [latex]f^{\\prime \\prime}[\/latex] is continuous over [latex]I[\/latex] and remains positive, [latex]f[\/latex] is consistently concave up across [latex]I[\/latex], which helps in determining the behavior of [latex]f[\/latex] at other critical points.<\/p>\r\n<p>For instance, suppose there exists a point [latex]b[\/latex] such that [latex]f^{\\prime}(b)=0[\/latex] and [latex]f^{\\prime \\prime}[\/latex] is positive throughout, [latex]f[\/latex] has a local minimum at [latex]b[\/latex]. The second derivative thus confirms the nature of local extrema by providing insight into the concavity of the function at critical points.<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img id=\"21\" src=\"https:\/\/openstax.org\/apps\/archive\/20210421.141058\/resources\/b4c5748b2ddf90fa36c58b811e8bda2a1d0025e5\" alt=\"A function f(x) is graphed in the first quadrant with a and b marked on the x-axis. The function is vaguely sinusoidal, increasing first to x = a, then decreasing to x = b, and increasing again. At (a, f(a)), the tangent is marked, and it is noted that f\u2019(a) = 0 and f\u2019\u2019(a) &lt; 0. At (b, f(b)), the tangent is marked, and it is noted f\u2019(b) = 0 and f\u2019\u2019(b) &gt; 0.\" width=\"487\" height=\"272\" \/> Figure 9. Consider a twice-differentiable function [latex]f[\/latex] such that [latex]f^{\\prime \\prime}[\/latex] is continuous. Since [latex]f^{\\prime}(a)=0[\/latex] and [latex]f^{\\prime \\prime}(a)&lt;0[\/latex], there is an interval [latex]I[\/latex] containing [latex]a[\/latex] such that for all [latex]x[\/latex] in [latex]I[\/latex], [latex]f[\/latex] is increasing if [latex]x&lt;a[\/latex] and [latex]f[\/latex] is decreasing if [latex]x&gt;a[\/latex]. As a result, [latex]f[\/latex] has a local maximum at [latex]x=a[\/latex]. Since [latex]f^{\\prime}(b)=0[\/latex] and [latex]f^{\\prime \\prime}(b)&gt;0[\/latex], there is an interval [latex]I[\/latex] containing [latex]b[\/latex] such that for all [latex]x[\/latex] in [latex]I[\/latex], [latex]f[\/latex] is decreasing if [latex]x&lt;b[\/latex] and [latex]f[\/latex] is increasing if [latex]x&gt;b[\/latex]. As a result, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex].[\/caption]\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3 style=\"text-align: left;\">second derivative test<\/h3>\r\n<p id=\"fs-id1165043254229\">Suppose [latex]f^{\\prime}(c)=0, \\, f^{\\prime \\prime}[\/latex] is continuous over an interval containing [latex]c[\/latex].<\/p>\r\n<ol id=\"fs-id1165042319133\" style=\"list-style-type: lower-roman;\">\r\n\t<li>If [latex]f^{\\prime \\prime}(c)&gt;0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime \\prime}(c)&lt;0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime \\prime}(c)=0[\/latex], then the test is inconclusive.<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>Note that for case iii. when [latex]f^{\\prime \\prime}(c)=0[\/latex], then [latex]f[\/latex] may have a local maximum, local minimum, or neither at [latex]c[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>The functions [latex]f(x)=x^3[\/latex], [latex]f(x)=x^4[\/latex], and [latex]f(x)=\u2212x^4[\/latex] all have critical points at [latex]x=0[\/latex]. In each case, the second derivative is zero at [latex]x=0[\/latex].<\/p>\r\n<p>However, the function [latex]f(x)=x^4[\/latex] has a local minimum at [latex]x=0[\/latex] whereas the function [latex]f(x)=\u2212x^4[\/latex] has a local maximum at [latex]x=0[\/latex] and the function [latex]f(x)=x^3[\/latex] does not have a local extremum at [latex]x=0[\/latex].<\/p>\r\n<\/section>\r\n<p id=\"fs-id1165043425387\">Let\u2019s now look at how to use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at a critical point [latex]c[\/latex] where [latex]f^{\\prime}(c)=0[\/latex].<\/p>\r\n<section class=\"textbox example\">\r\n<p>Use the second derivative to find the location of all local extrema for [latex]f(x)=x^5-5x^3[\/latex].<\/p>\r\n\r\n[reveal-answer q=\"fs-id1165042320876\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042320876\"]\r\n\r\n<p id=\"fs-id1165042320876\">To apply the second derivative test, we first need to find critical points [latex]c[\/latex] where [latex]f^{\\prime}(c)=0[\/latex].<\/p>\r\n<p>The derivative is [latex]f^{\\prime}(x)=5x^4-15x^2[\/latex]. Therefore, [latex]f^{\\prime}(x)=5x^4-15x^2=5x^2(x^2-3)=0[\/latex] when [latex]x=0,\\pm \\sqrt{3}[\/latex].<\/p>\r\n<p id=\"fs-id1165043431475\">To determine whether [latex]f[\/latex] has a local extrema at any of these points, we need to evaluate the sign of [latex]f^{\\prime \\prime}[\/latex] at these points. The second derivative is<\/p>\r\n<div id=\"fs-id1165042318723\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime \\prime}(x)=20x^3-30x=10x(2x^2-3)[\/latex].<\/div>\r\n<p id=\"fs-id1165042373916\">In the following table, we evaluate the second derivative at each of the critical points and use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at any of these points.<\/p>\r\n<table id=\"fs-id1165042705774\" class=\"unnumbered\" summary=\"This table has four rows and three columns. The first row is a header row, and it reads x, f\u2019\u2019(x), and Conclusion. After the header, the first column reads negative square root of 3, 0, and square root of 3. The second column reads negative 30 times the square root of 3, 0, and 30 times the square root of 3. The third column reads Local maxiumum, Second derivative test is inconclusive, and Local minimum.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>[latex]x[\/latex]<\/th>\r\n<th>[latex]f^{\\prime \\prime}(x)[\/latex]<\/th>\r\n<th>Conclusion<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>[latex]\u2212\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]-30\\sqrt{3}[\/latex]<\/td>\r\n<td>Local maximum<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>Second derivative test is inconclusive<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td>[latex]30\\sqrt{3}[\/latex]<\/td>\r\n<td>Local minimum<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042476082\">By the second derivative test, we conclude that [latex]f[\/latex] has a local maximum at [latex]x=\u2212\\sqrt{3}[\/latex] and [latex]f[\/latex] has a local minimum at [latex]x=\\sqrt{3}[\/latex]. The second derivative test is inconclusive at [latex]x=0[\/latex].<\/p>\r\n<p>To determine whether [latex]f[\/latex] has a local extrema at [latex]x=0[\/latex], we apply the first derivative test.<\/p>\r\n<p>To evaluate the sign of [latex]f^{\\prime}(x)=5x^2(x^2-3)[\/latex] for [latex]x \\in (\u2212\\sqrt{3},0)[\/latex] and [latex]x \\in (0,\\sqrt{3})[\/latex], let [latex]x=-1[\/latex] and [latex]x=1[\/latex] be the two test points. Since [latex]f^{\\prime}(-1)&lt;0[\/latex] and [latex]f^{\\prime}(1)&lt;0[\/latex], we conclude that [latex]f[\/latex] is decreasing on both intervals and, therefore, [latex]f[\/latex] does not have a local extrema at [latex]x=0[\/latex] as shown in the following graph.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"801\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210942\/CNX_Calc_Figure_04_05_007.jpg\" alt=\"The function f(x) = x5 \u2013 5x3 is graphed. The function increases to (negative square root of 3, 10), then decreases to an inflection point at 0, continues decreasing to (square root of 3, \u221210), and then increases.\" width=\"801\" height=\"408\" \/> Figure 10. The function [latex]f[\/latex] has a local maximum at [latex]x=\u2212\\sqrt{3}[\/latex] and a local minimum at [latex]x=\\sqrt{3}[\/latex][\/caption]\r\n\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/PBeo4ZJ-FGY?controls=0&amp;start=789&amp;end=1016&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a style=\"font-size: 1rem; orphans: 1; text-align: initial;\" href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.5DerivativesAndTheShapeOfAGraph789to1016_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.5 Derivatives and the Shape of a Graph\" here (opens in new window)<\/a><\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question]209260[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<h2>The Second Derivative Test<\/h2>\n<p>The first derivative test provides a systematic approach to identify local extrema, but in some cases, using the second derivative can be more straightforward. A function must have a local extremum at a critical point, but not all critical points are extremas.<\/p>\n<p>Consider a function [latex]f[\/latex] that is twice-differentiable on an open interval [latex]I[\/latex] containing [latex]a[\/latex].<\/p>\n<ul>\n<li>If [latex]f^{\\prime \\prime}(x)<0[\/latex] and [latex]f^{\\prime}(a)=0[\/latex], [latex]f[\/latex]is concave down at [latex]a[\/latex], indicating a local maximum.<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)>0[\/latex] and [latex]f^{\\prime}(a)=0[\/latex], [latex]f[\/latex]is concave up at [latex]a[\/latex], suggesting a local minimum at [latex]a[\/latex].<\/li>\n<\/ul>\n<p>Furthermore, if [latex]f^{\\prime \\prime}[\/latex] is continuous over [latex]I[\/latex] and remains positive, [latex]f[\/latex] is consistently concave up across [latex]I[\/latex], which helps in determining the behavior of [latex]f[\/latex] at other critical points.<\/p>\n<p>For instance, suppose there exists a point [latex]b[\/latex] such that [latex]f^{\\prime}(b)=0[\/latex] and [latex]f^{\\prime \\prime}[\/latex] is positive throughout, [latex]f[\/latex] has a local minimum at [latex]b[\/latex]. The second derivative thus confirms the nature of local extrema by providing insight into the concavity of the function at critical points.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" id=\"21\" src=\"https:\/\/openstax.org\/apps\/archive\/20210421.141058\/resources\/b4c5748b2ddf90fa36c58b811e8bda2a1d0025e5\" alt=\"A function f(x) is graphed in the first quadrant with a and b marked on the x-axis. The function is vaguely sinusoidal, increasing first to x = a, then decreasing to x = b, and increasing again. At (a, f(a)), the tangent is marked, and it is noted that f\u2019(a) = 0 and f\u2019\u2019(a) &lt; 0. At (b, f(b)), the tangent is marked, and it is noted f\u2019(b) = 0 and f\u2019\u2019(b) &gt; 0.\" width=\"487\" height=\"272\" \/><figcaption class=\"wp-caption-text\">Figure 9. Consider a twice-differentiable function [latex]f[\/latex] such that [latex]f^{\\prime \\prime}[\/latex] is continuous. Since [latex]f^{\\prime}(a)=0[\/latex] and [latex]f^{\\prime \\prime}(a)&lt;0[\/latex], there is an interval [latex]I[\/latex] containing [latex]a[\/latex] such that for all [latex]x[\/latex] in [latex]I[\/latex], [latex]f[\/latex] is increasing if [latex]x&lt;a[\/latex] and [latex]f[\/latex] is decreasing if [latex]x&gt;a[\/latex]. As a result, [latex]f[\/latex] has a local maximum at [latex]x=a[\/latex]. Since [latex]f^{\\prime}(b)=0[\/latex] and [latex]f^{\\prime \\prime}(b)&gt;0[\/latex], there is an interval [latex]I[\/latex] containing [latex]b[\/latex] such that for all [latex]x[\/latex] in [latex]I[\/latex], [latex]f[\/latex] is decreasing if [latex]x&lt;b[\/latex] and [latex]f[\/latex] is increasing if [latex]x&gt;b[\/latex]. As a result, [latex]f[\/latex] has a local minimum at [latex]x=b[\/latex].<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">second derivative test<\/h3>\n<p id=\"fs-id1165043254229\">Suppose [latex]f^{\\prime}(c)=0, \\, f^{\\prime \\prime}[\/latex] is continuous over an interval containing [latex]c[\/latex].<\/p>\n<ol id=\"fs-id1165042319133\" style=\"list-style-type: lower-roman;\">\n<li>If [latex]f^{\\prime \\prime}(c)>0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(c)<0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(c)=0[\/latex], then the test is inconclusive.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox proTip\">\n<p>Note that for case iii. when [latex]f^{\\prime \\prime}(c)=0[\/latex], then [latex]f[\/latex] may have a local maximum, local minimum, or neither at [latex]c[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>The functions [latex]f(x)=x^3[\/latex], [latex]f(x)=x^4[\/latex], and [latex]f(x)=\u2212x^4[\/latex] all have critical points at [latex]x=0[\/latex]. In each case, the second derivative is zero at [latex]x=0[\/latex].<\/p>\n<p>However, the function [latex]f(x)=x^4[\/latex] has a local minimum at [latex]x=0[\/latex] whereas the function [latex]f(x)=\u2212x^4[\/latex] has a local maximum at [latex]x=0[\/latex] and the function [latex]f(x)=x^3[\/latex] does not have a local extremum at [latex]x=0[\/latex].<\/p>\n<\/section>\n<p id=\"fs-id1165043425387\">Let\u2019s now look at how to use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at a critical point [latex]c[\/latex] where [latex]f^{\\prime}(c)=0[\/latex].<\/p>\n<section class=\"textbox example\">\n<p>Use the second derivative to find the location of all local extrema for [latex]f(x)=x^5-5x^3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042320876\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042320876\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042320876\">To apply the second derivative test, we first need to find critical points [latex]c[\/latex] where [latex]f^{\\prime}(c)=0[\/latex].<\/p>\n<p>The derivative is [latex]f^{\\prime}(x)=5x^4-15x^2[\/latex]. Therefore, [latex]f^{\\prime}(x)=5x^4-15x^2=5x^2(x^2-3)=0[\/latex] when [latex]x=0,\\pm \\sqrt{3}[\/latex].<\/p>\n<p id=\"fs-id1165043431475\">To determine whether [latex]f[\/latex] has a local extrema at any of these points, we need to evaluate the sign of [latex]f^{\\prime \\prime}[\/latex] at these points. The second derivative is<\/p>\n<div id=\"fs-id1165042318723\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime \\prime}(x)=20x^3-30x=10x(2x^2-3)[\/latex].<\/div>\n<p id=\"fs-id1165042373916\">In the following table, we evaluate the second derivative at each of the critical points and use the second derivative test to determine whether [latex]f[\/latex] has a local maximum or local minimum at any of these points.<\/p>\n<table id=\"fs-id1165042705774\" class=\"unnumbered\" summary=\"This table has four rows and three columns. The first row is a header row, and it reads x, f\u2019\u2019(x), and Conclusion. After the header, the first column reads negative square root of 3, 0, and square root of 3. The second column reads negative 30 times the square root of 3, 0, and 30 times the square root of 3. The third column reads Local maxiumum, Second derivative test is inconclusive, and Local minimum.\">\n<thead>\n<tr valign=\"top\">\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f^{\\prime \\prime}(x)[\/latex]<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex]\u2212\\sqrt{3}[\/latex]<\/td>\n<td>[latex]-30\\sqrt{3}[\/latex]<\/td>\n<td>Local maximum<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>Second derivative test is inconclusive<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex]\\sqrt{3}[\/latex]<\/td>\n<td>[latex]30\\sqrt{3}[\/latex]<\/td>\n<td>Local minimum<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042476082\">By the second derivative test, we conclude that [latex]f[\/latex] has a local maximum at [latex]x=\u2212\\sqrt{3}[\/latex] and [latex]f[\/latex] has a local minimum at [latex]x=\\sqrt{3}[\/latex]. The second derivative test is inconclusive at [latex]x=0[\/latex].<\/p>\n<p>To determine whether [latex]f[\/latex] has a local extrema at [latex]x=0[\/latex], we apply the first derivative test.<\/p>\n<p>To evaluate the sign of [latex]f^{\\prime}(x)=5x^2(x^2-3)[\/latex] for [latex]x \\in (\u2212\\sqrt{3},0)[\/latex] and [latex]x \\in (0,\\sqrt{3})[\/latex], let [latex]x=-1[\/latex] and [latex]x=1[\/latex] be the two test points. Since [latex]f^{\\prime}(-1)<0[\/latex] and [latex]f^{\\prime}(1)<0[\/latex], we conclude that [latex]f[\/latex] is decreasing on both intervals and, therefore, [latex]f[\/latex] does not have a local extrema at [latex]x=0[\/latex] as shown in the following graph.<\/p>\n<figure style=\"width: 801px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11210942\/CNX_Calc_Figure_04_05_007.jpg\" alt=\"The function f(x) = x5 \u2013 5x3 is graphed. The function increases to (negative square root of 3, 10), then decreases to an inflection point at 0, continues decreasing to (square root of 3, \u221210), and then increases.\" width=\"801\" height=\"408\" \/><figcaption class=\"wp-caption-text\">Figure 10. The function [latex]f[\/latex] has a local maximum at [latex]x=\u2212\\sqrt{3}[\/latex] and a local minimum at [latex]x=\\sqrt{3}[\/latex]<\/figcaption><\/figure>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/PBeo4ZJ-FGY?controls=0&amp;start=789&amp;end=1016&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a style=\"font-size: 1rem; orphans: 1; text-align: initial;\" href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.5DerivativesAndTheShapeOfAGraph789to1016_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.5 Derivatives and the Shape of a Graph&#8221; here (opens in new window)<\/a><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm209260\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=209260&theme=lumen&iframe_resize_id=ohm209260&source=tnh&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":27,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"4.5 Derivatives and the Shape of a Graph\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":652,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"4.5 Derivatives and the Shape of a Graph","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"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\/309"}],"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\/6"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/309\/revisions"}],"predecessor-version":[{"id":4344,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/309\/revisions\/4344"}],"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\/309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}