{"id":2284,"date":"2021-09-21T15:11:49","date_gmt":"2021-09-21T15:11:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-the-divergence-and-integral-tests\/"},"modified":"2022-04-19T20:47:41","modified_gmt":"2022-04-19T20:47:41","slug":"skills-review-for-the-divergence-and-integral-tests","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-the-divergence-and-integral-tests\/","title":{"raw":"Skills Review for The Divergence and Integral Tests","rendered":"Skills Review for The Divergence and Integral Tests"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Calculate the limit of a function as \ud835\udc65 increases or decreases without bound<\/li>\r\n \t<li>Recognize when to apply L\u2019H\u00f4pital\u2019s rule<\/li>\r\n \t<li>Explain how the sign of the first derivative affects the shape of a function\u2019s graph<\/li>\r\n \t<li>State the first derivative test for critical points<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Divergence and Integral Tests section, we will explore some methods that can be used to determine whether an infinite series diverges or converges. Here we will review how to take limits at infinity, L'Hopital's Rule, and how to determine where a function is decreasing and increasing.\r\n<h2>Take Limits at Infinity<\/h2>\r\n<em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em>\r\n<h2>Infinite Limits at Infinity<\/h2>\r\n<em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em>\r\n<h2>Apply L'H\u00f4pital's Rule<\/h2>\r\n<em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em>\r\n<div id=\"fs-id1165043199295\" class=\"bc-section section\">\r\n<h2>The First Derivative Test<\/h2>\r\n<p id=\"fs-id1165043093996\">If the derivative of a function is positive over an interval [latex]I[\/latex] then the function is increasing over [latex]I[\/latex]. On the other hand, if the derivative of the function is negative over an interval [latex]I[\/latex], then the function is decreasing over [latex]I[\/latex] as shown in the following figure.<\/p>\r\n<img id=\"2\" src=\"https:\/\/openstax.org\/resources\/9f1308c124662c7cc7ef9debf1112c9c47e96df6\" alt=\"&quot;This\" \/>\r\n\r\nRecall that\u00a0[latex]c[\/latex] is a <strong>critical point<\/strong> of a function [latex]f[\/latex] if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined.\r\n<ul id=\"fs-id1165043257290\">\r\n \t<li>If a continuous function [latex]f[\/latex] has a local extremum, it must occur at a critical point [latex]c[\/latex].<\/li>\r\n \t<li>The function has a local extremum at the critical point [latex]c[\/latex] if and only if the derivative [latex]f^{\\prime}[\/latex] switches sign as [latex]x[\/latex] increases through [latex]c[\/latex].<\/li>\r\n \t<li>Therefore, to test whether a function has a local extremum at a critical point [latex]c[\/latex], we must determine the sign of [latex]f^{\\prime}(x)[\/latex] to the left and right of [latex]c[\/latex].<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165043262520\">This result is known as the <strong>first derivative test<\/strong>.<\/p>\r\n\r\n<div id=\"fs-id1165042327970\" class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">First Derivative Test<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1165043067802\">Suppose that [latex]f[\/latex] is a continuous function over an interval [latex]I[\/latex] containing a critical point [latex]c[\/latex]. If [latex]f[\/latex] is differentiable over [latex]I[\/latex], except possibly at point [latex]c[\/latex], then [latex]f(c)[\/latex] satisfies one of the following descriptions:<\/p>\r\n\r\n<ol id=\"fs-id1165043068012\">\r\n \t<li>If [latex]f^{\\prime}[\/latex] changes sign from positive when [latex]x&lt;c[\/latex] to negative when [latex]x&gt;c[\/latex], then [latex]f(c)[\/latex] is a local maximum of [latex]f[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime}[\/latex] changes sign from negative when [latex]x&lt;c[\/latex] to positive when [latex]x&gt;c[\/latex], then [latex]f(c)[\/latex] is a local minimum of [latex]f[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime}[\/latex] has the same sign for [latex]x&lt;c[\/latex] and [latex]x&gt;c[\/latex], then [latex]f(c)[\/latex] is neither a local maximum nor a local minimum of [latex]f[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165043423022\" class=\"textbook exercises\">\r\n<h3>Example: Using the First Derivative Test to Find Increasing And Decreasing Intervals<\/h3>\r\nUse the first derivative test to find all increasing and decreasing intervals for [latex]f(x)=x^3-3x^2-9x-1[\/latex].\r\n<div id=\"fs-id1165042973802\" class=\"exercise\">[reveal-answer q=\"fs-id1165043253635\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043253635\"]\r\n<p id=\"fs-id1165043253635\">The derivative is [latex]f^{\\prime}(x)=3x^2-6x-9[\/latex]. To find the critical points, we need to find where [latex]f^{\\prime}(x)=0[\/latex]. Factoring the polynomial, we conclude that the critical points must satisfy<\/p>\r\n\r\n<div id=\"fs-id1165043318988\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]3(x^2-2x-3)=3(x-3)(x+1)=0[\/latex]<\/div>\r\n<p id=\"fs-id1165043041063\">Therefore, the critical points are [latex]x=3,-1[\/latex]. Now divide the interval [latex](\u2212\\infty ,\\infty)[\/latex] into the smaller intervals [latex](\u2212\\infty ,-1), \\, (-1,3)[\/latex], and [latex](3,\\infty)[\/latex].<\/p>\r\n<p id=\"fs-id1165043097583\">Since [latex]f^{\\prime}[\/latex] is a continuous function, to determine the sign of [latex]f^{\\prime}(x)[\/latex] over each subinterval, it suffices to choose a point over each of the intervals [latex](\u2212\\infty ,-1), \\, (-1,3)[\/latex], and [latex](3,\\infty)[\/latex] and determine the sign of [latex]f^{\\prime}[\/latex] at each of these points. For example, let\u2019s choose [latex]x=-2, \\, x=0[\/latex], and [latex]x=4[\/latex] as test points.<\/p>\r\n\r\n<table id=\"fs-id1165042978447\" class=\"unnumbered\" summary=\"This table has four rows and four columns. The first row is a header row, and it reads from left to right Interval, Test Point, Sign of f\u2019(x) = 3(x \u22123)(x + 1) at Test Point, and Conclusion. Below the header, the first column reads (\u2212\u221e, \u22121), (\u22121, 3), and (3, \u221e). The second column reads x = \u22122, x = 0, and x = 4. The third column reads (+)(\u2212)(\u2212) = +, (+)(\u2212)(+) = \u2212, and (+)(+)(+) = +. The fourth column reads f is increasing, f is decreasing, and f is increasing.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]f^{\\prime}(x)=3(x-3)(x+1)[\/latex] at Test Point<\/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\\infty ,-1)[\/latex]<\/td>\r\n<td>[latex]x=-2[\/latex]<\/td>\r\n<td>[latex](+)(\u2212)(\u2212)=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](-1,3)[\/latex]<\/td>\r\n<td>[latex]x=0[\/latex]<\/td>\r\n<td>[latex](+)(\u2212)(+)=\u2212[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](3,\\infty)[\/latex]<\/td>\r\n<td>[latex]x=4[\/latex]<\/td>\r\n<td>[latex](+)(+)(+)=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165042931764\"><span style=\"font-size: 1em;\">[\/hidden-answer]<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbook key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse the first derivative test to determine the increasing and decreasing intervals for [latex]f(x)=\u2212x^3+\\frac{3}{2}x^2+18x[\/latex].\r\n\r\n[reveal-answer q=\"6192003\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"6192003\"]\r\n\r\nFind 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.\r\n\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1165043281485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043281485\"]\r\n\r\n[latex]f[\/latex] is decreasing on the intervals [latex](-\\infty, -2) and (3,\\infty)[\/latex] and increasing on the interval [latex](-2,3)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165043312782\" class=\"textbook exercises\">\r\n<h3>Example: Using the First Derivative Test to Find Increasing And Decreasing Intervals<\/h3>\r\nUse the first derivative test to find the increasing and decreasing intervals for [latex]f(x)=5x^{\\frac{1}{3}}-x^{\\frac{5}{3}}[\/latex]. Use a graphing utility to confirm your results.\r\n<div id=\"fs-id1165042640287\" class=\"exercise\">[reveal-answer q=\"fs-id1165043109175\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165043109175\"]\r\n<p id=\"fs-id1165043109175\">The derivative is<\/p>\r\n\r\n<div id=\"fs-id1165043066545\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\frac{5}{3}x^{-2\/3}-\\frac{5}{3}x^{2\/3}=\\frac{5}{3x^{2\/3}}-\\frac{5x^{2\/3}}{3}=\\frac{5-5x^{4\/3}}{3x^{2\/3}}=\\frac{5(1-x^{4\/3})}{3x^{2\/3}}[\/latex].<\/div>\r\nThe derivative [latex]f^{\\prime}(x)=0[\/latex] when [latex]1-x^{4\/3}=0[\/latex]. Therefore, [latex]f^{\\prime}(x)=0[\/latex] at [latex]x=\\pm 1[\/latex]. The derivative [latex]f^{\\prime}(x)[\/latex] is undefined at [latex]x=0[\/latex]. Therefore, we have three critical points: [latex]x=0[\/latex], [latex]x=1[\/latex], and [latex]x=-1[\/latex]. Consequently, divide the interval [latex](\u2212\\infty ,\\infty)[\/latex] into the smaller intervals [latex](\u2212\\infty ,-1), \\, (-1,0), \\, (0,1)[\/latex], and [latex](1,\\infty )[\/latex].\r\n<p id=\"fs-id1165042354821\">Since [latex]f^{\\prime}[\/latex] is continuous over each subinterval, it suffices to choose a test point [latex]x[\/latex] in each of the intervals from step 1 and determine the sign of [latex]f^{\\prime}[\/latex] at each of these points. The points [latex]x=-2, \\, x=-\\frac{1}{2}, \\, x=\\frac{1}{2}[\/latex], and [latex]x=2[\/latex] are test points for these intervals.<\/p>\r\n\r\n<table id=\"fs-id1165043032734\" class=\"unnumbered\" summary=\"This table has five rows and four columns. The first row is a header row, and it reads from left to right Interval, Test Point, Sign of f\u2019(x) = 5(1 \u2013 x4\/3)\/(3x2\/3) at Test Point, and Conclusion. Below the header, the first column reads (\u2212\u221e, \u22121), (\u22121, 0), (0, 1), and (1, \u221e). The second column reads x = \u22122, x = \u22121\/2, x = 1\/2, and x = 2. The third column reads (+)(\u2212)\/(+) = \u2212, (+)(+)\/(+) = +, (+)(+)\/(+) = +, and (+)(\u2212)\/(+) = \u2212. The fourth column reads f is decreasing, f is increasing, f is increasing, and f is decreasing.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Interval<\/th>\r\n<th>Test Point<\/th>\r\n<th>Sign of [latex]f^{\\prime}(x)=\\frac{5(1-x^{4\/3})}{3x^{2\/3}}[\/latex] at Test Point<\/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\\infty ,-1)[\/latex]<\/td>\r\n<td>[latex]x=-2[\/latex]<\/td>\r\n<td>[latex]\\frac{(+)(\u2212)}{+}=\u2212[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](-1,0)[\/latex]<\/td>\r\n<td>[latex]x=-\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{(+)(+)}{+}=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](0,1)[\/latex]<\/td>\r\n<td>[latex]x=\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{(+)(+)}{+}=+[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is increasing.<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>[latex](1,\\infty )[\/latex]<\/td>\r\n<td>[latex]x=2[\/latex]<\/td>\r\n<td>[latex]\\frac{(+)(\u2212)}{+}=\u2212[\/latex]<\/td>\r\n<td>[latex]f[\/latex] is decreasing.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]20222[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate the limit of a function as \ud835\udc65 increases or decreases without bound<\/li>\n<li>Recognize when to apply L\u2019H\u00f4pital\u2019s rule<\/li>\n<li>Explain how the sign of the first derivative affects the shape of a function\u2019s graph<\/li>\n<li>State the first derivative test for critical points<\/li>\n<\/ul>\n<\/div>\n<p>In the Divergence and Integral Tests section, we will explore some methods that can be used to determine whether an infinite series diverges or converges. Here we will review how to take limits at infinity, L&#8217;Hopital&#8217;s Rule, and how to determine where a function is decreasing and increasing.<\/p>\n<h2>Take Limits at Infinity<\/h2>\n<p><em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em><\/p>\n<h2>Infinite Limits at Infinity<\/h2>\n<p><em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em><\/p>\n<h2>Apply L&#8217;H\u00f4pital&#8217;s Rule<\/h2>\n<p><em><strong>(see <a href=\"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-sequences\/\" target=\"_blank\" rel=\"noopener\">Module 5, Skills Review for Sequences<\/a>.)<\/strong><\/em><\/p>\n<div id=\"fs-id1165043199295\" class=\"bc-section section\">\n<h2>The First Derivative Test<\/h2>\n<p id=\"fs-id1165043093996\">If the derivative of a function is positive over an interval [latex]I[\/latex] then the function is increasing over [latex]I[\/latex]. On the other hand, if the derivative of the function is negative over an interval [latex]I[\/latex], then the function is decreasing over [latex]I[\/latex] as shown in the following figure.<\/p>\n<p><img decoding=\"async\" id=\"2\" src=\"https:\/\/openstax.org\/resources\/9f1308c124662c7cc7ef9debf1112c9c47e96df6\" alt=\"&quot;This\" \/><\/p>\n<p>Recall that\u00a0[latex]c[\/latex] is a <strong>critical point<\/strong> of a function [latex]f[\/latex] if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined.<\/p>\n<ul id=\"fs-id1165043257290\">\n<li>If a continuous function [latex]f[\/latex] has a local extremum, it must occur at a critical point [latex]c[\/latex].<\/li>\n<li>The function has a local extremum at the critical point [latex]c[\/latex] if and only if the derivative [latex]f^{\\prime}[\/latex] switches sign as [latex]x[\/latex] increases through [latex]c[\/latex].<\/li>\n<li>Therefore, to test whether a function has a local extremum at a critical point [latex]c[\/latex], we must determine the sign of [latex]f^{\\prime}(x)[\/latex] to the left and right of [latex]c[\/latex].<\/li>\n<\/ul>\n<p id=\"fs-id1165043262520\">This result is known as the <strong>first derivative test<\/strong>.<\/p>\n<div id=\"fs-id1165042327970\" class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">First Derivative Test<\/h3>\n<hr \/>\n<p id=\"fs-id1165043067802\">Suppose that [latex]f[\/latex] is a continuous function over an interval [latex]I[\/latex] containing a critical point [latex]c[\/latex]. If [latex]f[\/latex] is differentiable over [latex]I[\/latex], except possibly at point [latex]c[\/latex], then [latex]f(c)[\/latex] satisfies one of the following descriptions:<\/p>\n<ol id=\"fs-id1165043068012\">\n<li>If [latex]f^{\\prime}[\/latex] changes sign from positive when [latex]x<c[\/latex] to negative when [latex]x>c[\/latex], then [latex]f(c)[\/latex] is a local maximum of [latex]f[\/latex].<\/li>\n<li>If [latex]f^{\\prime}[\/latex] changes sign from negative when [latex]x<c[\/latex] to positive when [latex]x>c[\/latex], then [latex]f(c)[\/latex] is a local minimum of [latex]f[\/latex].<\/li>\n<li>If [latex]f^{\\prime}[\/latex] has the same sign for [latex]x<c[\/latex] and [latex]x>c[\/latex], then [latex]f(c)[\/latex] is neither a local maximum nor a local minimum of [latex]f[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165043423022\" class=\"textbook exercises\">\n<h3>Example: Using the First Derivative Test to Find Increasing And Decreasing Intervals<\/h3>\n<p>Use the first derivative test to find all increasing and decreasing intervals for [latex]f(x)=x^3-3x^2-9x-1[\/latex].<\/p>\n<div id=\"fs-id1165042973802\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043253635\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043253635\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043253635\">The derivative is [latex]f^{\\prime}(x)=3x^2-6x-9[\/latex]. To find the critical points, we need to find where [latex]f^{\\prime}(x)=0[\/latex]. Factoring the polynomial, we conclude that the critical points must satisfy<\/p>\n<div id=\"fs-id1165043318988\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]3(x^2-2x-3)=3(x-3)(x+1)=0[\/latex]<\/div>\n<p id=\"fs-id1165043041063\">Therefore, the critical points are [latex]x=3,-1[\/latex]. Now divide the interval [latex](\u2212\\infty ,\\infty)[\/latex] into the smaller intervals [latex](\u2212\\infty ,-1), \\, (-1,3)[\/latex], and [latex](3,\\infty)[\/latex].<\/p>\n<p id=\"fs-id1165043097583\">Since [latex]f^{\\prime}[\/latex] is a continuous function, to determine the sign of [latex]f^{\\prime}(x)[\/latex] over each subinterval, it suffices to choose a point over each of the intervals [latex](\u2212\\infty ,-1), \\, (-1,3)[\/latex], and [latex](3,\\infty)[\/latex] and determine the sign of [latex]f^{\\prime}[\/latex] at each of these points. For example, let\u2019s choose [latex]x=-2, \\, x=0[\/latex], and [latex]x=4[\/latex] as test points.<\/p>\n<table id=\"fs-id1165042978447\" class=\"unnumbered\" summary=\"This table has four rows and four columns. The first row is a header row, and it reads from left to right Interval, Test Point, Sign of f\u2019(x) = 3(x \u22123)(x + 1) at Test Point, and Conclusion. Below the header, the first column reads (\u2212\u221e, \u22121), (\u22121, 3), and (3, \u221e). The second column reads x = \u22122, x = 0, and x = 4. The third column reads (+)(\u2212)(\u2212) = +, (+)(\u2212)(+) = \u2212, and (+)(+)(+) = +. The fourth column reads f is increasing, f is decreasing, and f is increasing.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]f^{\\prime}(x)=3(x-3)(x+1)[\/latex] at Test Point<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\u2212\\infty ,-1)[\/latex]<\/td>\n<td>[latex]x=-2[\/latex]<\/td>\n<td>[latex](+)(\u2212)(\u2212)=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](-1,3)[\/latex]<\/td>\n<td>[latex]x=0[\/latex]<\/td>\n<td>[latex](+)(\u2212)(+)=\u2212[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](3,\\infty)[\/latex]<\/td>\n<td>[latex]x=4[\/latex]<\/td>\n<td>[latex](+)(+)(+)=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165042931764\"><span style=\"font-size: 1em;\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbook key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the first derivative test to determine the increasing and decreasing intervals for [latex]f(x)=\u2212x^3+\\frac{3}{2}x^2+18x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q6192003\">Hint<\/span><\/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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043281485\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043281485\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f[\/latex] is decreasing on the intervals [latex](-\\infty, -2) and (3,\\infty)[\/latex] and increasing on the interval [latex](-2,3)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165043312782\" class=\"textbook exercises\">\n<h3>Example: Using the First Derivative Test to Find Increasing And Decreasing Intervals<\/h3>\n<p>Use the first derivative test to find the increasing and decreasing intervals for [latex]f(x)=5x^{\\frac{1}{3}}-x^{\\frac{5}{3}}[\/latex]. Use a graphing utility to confirm your results.<\/p>\n<div id=\"fs-id1165042640287\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165043109175\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165043109175\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043109175\">The derivative is<\/p>\n<div id=\"fs-id1165043066545\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(x)=\\frac{5}{3}x^{-2\/3}-\\frac{5}{3}x^{2\/3}=\\frac{5}{3x^{2\/3}}-\\frac{5x^{2\/3}}{3}=\\frac{5-5x^{4\/3}}{3x^{2\/3}}=\\frac{5(1-x^{4\/3})}{3x^{2\/3}}[\/latex].<\/div>\n<p>The derivative [latex]f^{\\prime}(x)=0[\/latex] when [latex]1-x^{4\/3}=0[\/latex]. Therefore, [latex]f^{\\prime}(x)=0[\/latex] at [latex]x=\\pm 1[\/latex]. The derivative [latex]f^{\\prime}(x)[\/latex] is undefined at [latex]x=0[\/latex]. Therefore, we have three critical points: [latex]x=0[\/latex], [latex]x=1[\/latex], and [latex]x=-1[\/latex]. Consequently, divide the interval [latex](\u2212\\infty ,\\infty)[\/latex] into the smaller intervals [latex](\u2212\\infty ,-1), \\, (-1,0), \\, (0,1)[\/latex], and [latex](1,\\infty )[\/latex].<\/p>\n<p id=\"fs-id1165042354821\">Since [latex]f^{\\prime}[\/latex] is continuous over each subinterval, it suffices to choose a test point [latex]x[\/latex] in each of the intervals from step 1 and determine the sign of [latex]f^{\\prime}[\/latex] at each of these points. The points [latex]x=-2, \\, x=-\\frac{1}{2}, \\, x=\\frac{1}{2}[\/latex], and [latex]x=2[\/latex] are test points for these intervals.<\/p>\n<table id=\"fs-id1165043032734\" class=\"unnumbered\" summary=\"This table has five rows and four columns. The first row is a header row, and it reads from left to right Interval, Test Point, Sign of f\u2019(x) = 5(1 \u2013 x4\/3)\/(3x2\/3) at Test Point, and Conclusion. Below the header, the first column reads (\u2212\u221e, \u22121), (\u22121, 0), (0, 1), and (1, \u221e). The second column reads x = \u22122, x = \u22121\/2, x = 1\/2, and x = 2. The third column reads (+)(\u2212)\/(+) = \u2212, (+)(+)\/(+) = +, (+)(+)\/(+) = +, and (+)(\u2212)\/(+) = \u2212. The fourth column reads f is decreasing, f is increasing, f is increasing, and f is decreasing.\">\n<thead>\n<tr valign=\"top\">\n<th>Interval<\/th>\n<th>Test Point<\/th>\n<th>Sign of [latex]f^{\\prime}(x)=\\frac{5(1-x^{4\/3})}{3x^{2\/3}}[\/latex] at Test Point<\/th>\n<th>Conclusion<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td>[latex](\u2212\\infty ,-1)[\/latex]<\/td>\n<td>[latex]x=-2[\/latex]<\/td>\n<td>[latex]\\frac{(+)(\u2212)}{+}=\u2212[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](-1,0)[\/latex]<\/td>\n<td>[latex]x=-\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{(+)(+)}{+}=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](0,1)[\/latex]<\/td>\n<td>[latex]x=\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{(+)(+)}{+}=+[\/latex]<\/td>\n<td>[latex]f[\/latex] is increasing.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>[latex](1,\\infty )[\/latex]<\/td>\n<td>[latex]x=2[\/latex]<\/td>\n<td>[latex]\\frac{(+)(\u2212)}{+}=\u2212[\/latex]<\/td>\n<td>[latex]f[\/latex] is decreasing.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm20222\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=20222&theme=oea&iframe_resize_id=ohm20222&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2284\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen 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