{"id":2130,"date":"2021-03-27T17:07:04","date_gmt":"2021-03-27T17:07:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=2130"},"modified":"2021-04-03T02:16:45","modified_gmt":"2021-04-03T02:16:45","slug":"summary-of-derivatives-and-the-shape-of-a-graph","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-derivatives-and-the-shape-of-a-graph\/","title":{"raw":"Summary of Derivatives and the Shape of a Graph","rendered":"Summary of Derivatives and the Shape of a Graph"},"content":{"raw":"<div id=\"fs-id1165042606819\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165043163899\">\r\n \t<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)&gt;0[\/latex] for [latex]x&lt;c[\/latex] and [latex]f^{\\prime}(x)&lt;0[\/latex] for [latex]x&gt;c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\r\n \t<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)&lt;0[\/latex] for [latex]x&lt;c[\/latex] and [latex]f^{\\prime}(x)&gt;0[\/latex] for [latex]x&gt;c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime}(c)=0[\/latex] and [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}(c)=0[\/latex] and [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}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)=0[\/latex], then evaluate [latex]f^{\\prime}(x)[\/latex] at a test point [latex]x[\/latex] to the left of [latex]c[\/latex] and a test point [latex]x[\/latex] to the right of [latex]c[\/latex], to determine whether [latex]f[\/latex] has a local extremum at [latex]c[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\r\n \t<li>If [latex]f^{\\prime}(c)=0[\/latex] and [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}(c)=0[\/latex] and [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}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)=0[\/latex], then evaluate [latex]f^{\\prime}(x)[\/latex] at a test point [latex]x[\/latex] to the left of [latex]c[\/latex] and a test point [latex]x[\/latex] to the right of [latex]c[\/latex], to determine whether [latex]f[\/latex] has a local extremum at [latex]c[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165043427379\" class=\"definition\">\r\n \t<dt>concave down<\/dt>\r\n \t<dd id=\"fs-id1165043427385\">if [latex]f[\/latex] is differentiable over an interval [latex]I[\/latex] and [latex]f^{\\prime}[\/latex] is decreasing over [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043396223\" class=\"definition\">\r\n \t<dt>concave up<\/dt>\r\n \t<dd id=\"fs-id1165043396229\">if [latex]f[\/latex] is differentiable over an interval [latex]I[\/latex] and [latex]f^{\\prime}[\/latex] is increasing over [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043396264\" class=\"definition\">\r\n \t<dt>concavity<\/dt>\r\n \t<dd id=\"fs-id1165043396269\">the upward or downward curve of the graph of a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043396274\" class=\"definition\">\r\n \t<dt>concavity test<\/dt>\r\n \t<dd id=\"fs-id1165043396279\">suppose [latex]f[\/latex] is twice differentiable over an interval [latex]I[\/latex]; if [latex]f^{\\prime \\prime}&gt;0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex]; if [latex]f^{\\prime \\prime}&lt;0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043281605\" class=\"definition\">\r\n \t<dt>first derivative test<\/dt>\r\n \t<dd id=\"fs-id1165043281610\">let [latex]f[\/latex] be a continuous function over an interval [latex]I[\/latex] containing a critical point [latex]c[\/latex] such that [latex]f[\/latex] is differentiable over [latex]I[\/latex] except possibly at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] changes sign from positive to negative as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] changes sign from negative to positive as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] does not change sign as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] does not have a local extremum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042364543\" class=\"definition\">\r\n \t<dt>inflection point<\/dt>\r\n \t<dd id=\"fs-id1165042364548\">if [latex]f[\/latex] is continuous at [latex]c[\/latex] and [latex]f[\/latex] changes concavity at [latex]c[\/latex], the point [latex](c,f(c))[\/latex] is an inflection point of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043131579\" class=\"definition\">\r\n \t<dt>second derivative test<\/dt>\r\n \t<dd id=\"fs-id1165043131584\">suppose [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}[\/latex] is continuous over an interval containing [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)&gt;0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)&lt;0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)=0[\/latex], then the test is inconclusive<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1165042606819\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165043163899\">\n<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)>0[\/latex] for [latex]x<c[\/latex] and [latex]f^{\\prime}(x)<0[\/latex] for [latex]x>c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\n<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)<0[\/latex] for [latex]x<c[\/latex] and [latex]f^{\\prime}(x)>0[\/latex] for [latex]x>c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)>0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)<0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(c)=0[\/latex] and [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}(c)=0[\/latex] and [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}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)=0[\/latex], then evaluate [latex]f^{\\prime}(x)[\/latex] at a test point [latex]x[\/latex] to the left of [latex]c[\/latex] and a test point [latex]x[\/latex] to the right of [latex]c[\/latex], to determine whether [latex]f[\/latex] has a local extremum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)>0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)<0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(c)=0[\/latex] and [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}(c)=0[\/latex] and [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}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)=0[\/latex], then evaluate [latex]f^{\\prime}(x)[\/latex] at a test point [latex]x[\/latex] to the left of [latex]c[\/latex] and a test point [latex]x[\/latex] to the right of [latex]c[\/latex], to determine whether [latex]f[\/latex] has a local extremum at [latex]c[\/latex].<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165043427379\" class=\"definition\">\n<dt>concave down<\/dt>\n<dd id=\"fs-id1165043427385\">if [latex]f[\/latex] is differentiable over an interval [latex]I[\/latex] and [latex]f^{\\prime}[\/latex] is decreasing over [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043396223\" class=\"definition\">\n<dt>concave up<\/dt>\n<dd id=\"fs-id1165043396229\">if [latex]f[\/latex] is differentiable over an interval [latex]I[\/latex] and [latex]f^{\\prime}[\/latex] is increasing over [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043396264\" class=\"definition\">\n<dt>concavity<\/dt>\n<dd id=\"fs-id1165043396269\">the upward or downward curve of the graph of a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043396274\" class=\"definition\">\n<dt>concavity test<\/dt>\n<dd id=\"fs-id1165043396279\">suppose [latex]f[\/latex] is twice differentiable over an interval [latex]I[\/latex]; if [latex]f^{\\prime \\prime}>0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex]; if [latex]f^{\\prime \\prime}<0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043281605\" class=\"definition\">\n<dt>first derivative test<\/dt>\n<dd id=\"fs-id1165043281610\">let [latex]f[\/latex] be a continuous function over an interval [latex]I[\/latex] containing a critical point [latex]c[\/latex] such that [latex]f[\/latex] is differentiable over [latex]I[\/latex] except possibly at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] changes sign from positive to negative as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] changes sign from negative to positive as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] does not change sign as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] does not have a local extremum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042364543\" class=\"definition\">\n<dt>inflection point<\/dt>\n<dd id=\"fs-id1165042364548\">if [latex]f[\/latex] is continuous at [latex]c[\/latex] and [latex]f[\/latex] changes concavity at [latex]c[\/latex], the point [latex](c,f(c))[\/latex] is an inflection point of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043131579\" class=\"definition\">\n<dt>second derivative test<\/dt>\n<dd id=\"fs-id1165043131584\">suppose [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}[\/latex] is continuous over an interval containing [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)>0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)<0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)=0[\/latex], then the test is inconclusive<\/dd>\n<\/dl>\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-2130\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/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":17533,"menu_order":18,"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\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2130","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2130","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2130\/revisions"}],"predecessor-version":[{"id":2622,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2130\/revisions\/2622"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/48"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2130\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2130"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2130"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2130"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}