{"id":1638,"date":"2021-03-19T20:24:50","date_gmt":"2021-03-19T20:24:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1638"},"modified":"2021-03-27T18:07:53","modified_gmt":"2021-03-27T18:07:53","slug":"summary-of-maxima-and-minima","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-maxima-and-minima\/","title":{"raw":"Summary of Maxima and Minima","rendered":"Summary of Maxima and Minima"},"content":{"raw":"<div id=\"fs-id1165040729416\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1165040729423\">\r\n \t<li>A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.<\/li>\r\n \t<li>If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.<\/li>\r\n \t<li>A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165042059050\" class=\"definition\">\r\n \t<dt>absolute extremum<\/dt>\r\n \t<dd id=\"fs-id1165042059056\">if [latex]f[\/latex] has an absolute maximum or absolute minimum at [latex]c[\/latex], we say [latex]f[\/latex] has an absolute extremum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281356\" class=\"definition\">\r\n \t<dt>absolute maximum<\/dt>\r\n \t<dd id=\"fs-id1165042281361\">if [latex]f(c)\\ge f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], we say [latex]f[\/latex] has an absolute maximum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281408\" class=\"definition\">\r\n \t<dt>absolute minimum<\/dt>\r\n \t<dd id=\"fs-id1165042281414\">if [latex]f(c)\\le f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], we say [latex]f[\/latex] has an absolute minimum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281462\" class=\"definition\">\r\n \t<dt>critical point<\/dt>\r\n \t<dd id=\"fs-id1165042281467\">if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined, we say that [latex]c[\/latex] is a critical point of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281514\" class=\"definition\">\r\n \t<dt>extreme value theorem<\/dt>\r\n \t<dd id=\"fs-id1165042281519\">if [latex]f[\/latex] is a continuous function over a finite, closed interval, then [latex]f[\/latex] has an absolute maximum and an absolute minimum<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281532\" class=\"definition\">\r\n \t<dt>Fermat\u2019s theorem<\/dt>\r\n \t<dd id=\"fs-id1165042281538\">if [latex]f[\/latex] has a local extremum at [latex]c[\/latex], then [latex]c[\/latex] is a critical point of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281561\" class=\"definition\">\r\n \t<dt>local extremum<\/dt>\r\n \t<dd id=\"fs-id1165042281566\">if [latex]f[\/latex] has a local maximum or local minimum at [latex]c[\/latex], we say [latex]f[\/latex] has a local extremum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042071370\" class=\"definition\">\r\n \t<dt>local maximum<\/dt>\r\n \t<dd id=\"fs-id1165042071375\">if there exists an interval [latex]I[\/latex] such that [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex], we say [latex]f[\/latex] has a local maximum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042071428\" class=\"definition\">\r\n \t<dt>local minimum<\/dt>\r\n \t<dd id=\"fs-id1165042071434\">if there exists an interval [latex]I[\/latex] such that [latex]f(c)\\le f(x)[\/latex] for all [latex]x\\in I[\/latex], we say [latex]f[\/latex] has a local minimum at [latex]c[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1165040729416\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1165040729423\">\n<li>A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.<\/li>\n<li>If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.<\/li>\n<li>A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165042059050\" class=\"definition\">\n<dt>absolute extremum<\/dt>\n<dd id=\"fs-id1165042059056\">if [latex]f[\/latex] has an absolute maximum or absolute minimum at [latex]c[\/latex], we say [latex]f[\/latex] has an absolute extremum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281356\" class=\"definition\">\n<dt>absolute maximum<\/dt>\n<dd id=\"fs-id1165042281361\">if [latex]f(c)\\ge f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], we say [latex]f[\/latex] has an absolute maximum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281408\" class=\"definition\">\n<dt>absolute minimum<\/dt>\n<dd id=\"fs-id1165042281414\">if [latex]f(c)\\le f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], we say [latex]f[\/latex] has an absolute minimum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281462\" class=\"definition\">\n<dt>critical point<\/dt>\n<dd id=\"fs-id1165042281467\">if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined, we say that [latex]c[\/latex] is a critical point of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281514\" class=\"definition\">\n<dt>extreme value theorem<\/dt>\n<dd id=\"fs-id1165042281519\">if [latex]f[\/latex] is a continuous function over a finite, closed interval, then [latex]f[\/latex] has an absolute maximum and an absolute minimum<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281532\" class=\"definition\">\n<dt>Fermat\u2019s theorem<\/dt>\n<dd id=\"fs-id1165042281538\">if [latex]f[\/latex] has a local extremum at [latex]c[\/latex], then [latex]c[\/latex] is a critical point of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281561\" class=\"definition\">\n<dt>local extremum<\/dt>\n<dd id=\"fs-id1165042281566\">if [latex]f[\/latex] has a local maximum or local minimum at [latex]c[\/latex], we say [latex]f[\/latex] has a local extremum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042071370\" class=\"definition\">\n<dt>local maximum<\/dt>\n<dd id=\"fs-id1165042071375\">if there exists an interval [latex]I[\/latex] such that [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex], we say [latex]f[\/latex] has a local maximum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042071428\" class=\"definition\">\n<dt>local minimum<\/dt>\n<dd id=\"fs-id1165042071434\">if there exists an interval [latex]I[\/latex] such that [latex]f(c)\\le f(x)[\/latex] for all [latex]x\\in I[\/latex], we say [latex]f[\/latex] has a local minimum at [latex]c[\/latex]<\/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-1638\">\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":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) 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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-1638","chapter","type-chapter","status-publish","hentry"],"part":48,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1638","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":1,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1638\/revisions"}],"predecessor-version":[{"id":1639,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1638\/revisions\/1639"}],"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\/1638\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1638"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1638"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1638"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1638"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}