{"id":172,"date":"2021-07-30T17:24:34","date_gmt":"2021-07-30T17:24:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=172"},"modified":"2022-10-29T02:16:32","modified_gmt":"2022-10-29T02:16:32","slug":"summary-of-maxima-and-minima-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-maxima-and-minima-problems\/","title":{"raw":"Summary of Maxima and Minima Problems","rendered":"Summary of Maxima and Minima Problems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<div class=\"os-section-area\"><section id=\"fs-id1167793494963\" class=\"key-concepts\" data-depth=\"1\">\r\n<ul id=\"fs-id1167793494970\" data-bullet-style=\"bullet\">\r\n \t<li>A critical point of the function [latex]f(x,y)[\/latex] is any point\u00a0[latex](x_{0},y_{0})[\/latex] where either [latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[\/latex],\u00a0or at least one of\u00a0[latex]f_{x}(x_{0},y_{0})[\/latex] and\u00a0[latex]f_{y}(x_{0},y_{0})[\/latex] do not exist.<\/li>\r\n \t<li>A saddle point is a point\u00a0[latex](x_{0},y_{0})[\/latex] where\u00a0[latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[\/latex], but\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0is neither a maximum nor a minimum at that point.<\/li>\r\n \t<li>To find extrema of functions of two variables, first find the critical points, then calculate the discriminant and apply the second derivative test.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Discriminant<\/strong>\r\n[latex]D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\\left(f_{xy}(x_{0},y_{0})\\right)^{2}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>critical point of a function of two variables<\/dt>\r\n \t<dd>the point\u00a0[latex](x_{0},y_{0})[\/latex] is called a critical point of [latex]f(x,y)[\/latex] if one of the two following conditions holds:\r\n<ol>\r\n \t<li>[latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[\/latex]<\/li>\r\n \t<li>At least one of [latex]f_{x}(x_{0},y_{0})[\/latex] and [latex]f_{y}(x_{0},y_{0})[\/latex]<strong>\u00a0<\/strong>do not exist<\/li>\r\n<\/ol>\r\n<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>discriminant<\/dt>\r\n \t<dd>the discriminant of the function [latex]f(x,y)[\/latex]\u00a0is given by the formula\u00a0[latex]D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\\left(f_{xy}(x_{0},y_{0})\\right)^{2}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>saddle point<\/dt>\r\n \t<dd>given the function\u00a0[latex]z=f(x,y)[\/latex]\u00a0the point [latex](x_{0},y_{0},f(x_{0},y_{0}))[\/latex]\u00a0is a saddle point if both [latex]f_{x}(x_{0},y_{0})=0[\/latex] and [latex]f_{y}(x_{0},y_{0})=0[\/latex], but [latex]f[\/latex]\u00a0does not have a local extremum at\u00a0[latex](x_{0},y_{0})[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<div class=\"os-section-area\">\n<section id=\"fs-id1167793494963\" class=\"key-concepts\" data-depth=\"1\">\n<ul id=\"fs-id1167793494970\" data-bullet-style=\"bullet\">\n<li>A critical point of the function [latex]f(x,y)[\/latex] is any point\u00a0[latex](x_{0},y_{0})[\/latex] where either [latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[\/latex],\u00a0or at least one of\u00a0[latex]f_{x}(x_{0},y_{0})[\/latex] and\u00a0[latex]f_{y}(x_{0},y_{0})[\/latex] do not exist.<\/li>\n<li>A saddle point is a point\u00a0[latex](x_{0},y_{0})[\/latex] where\u00a0[latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[\/latex], but\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0is neither a maximum nor a minimum at that point.<\/li>\n<li>To find extrema of functions of two variables, first find the critical points, then calculate the discriminant and apply the second derivative test.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Discriminant<\/strong><br \/>\n[latex]D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\\left(f_{xy}(x_{0},y_{0})\\right)^{2}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>critical point of a function of two variables<\/dt>\n<dd>the point\u00a0[latex](x_{0},y_{0})[\/latex] is called a critical point of [latex]f(x,y)[\/latex] if one of the two following conditions holds:<\/p>\n<ol>\n<li>[latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[\/latex]<\/li>\n<li>At least one of [latex]f_{x}(x_{0},y_{0})[\/latex] and [latex]f_{y}(x_{0},y_{0})[\/latex]<strong>\u00a0<\/strong>do not exist<\/li>\n<\/ol>\n<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>discriminant<\/dt>\n<dd>the discriminant of the function [latex]f(x,y)[\/latex]\u00a0is given by the formula\u00a0[latex]D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\\left(f_{xy}(x_{0},y_{0})\\right)^{2}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>saddle point<\/dt>\n<dd>given the function\u00a0[latex]z=f(x,y)[\/latex]\u00a0the point [latex](x_{0},y_{0},f(x_{0},y_{0}))[\/latex]\u00a0is a saddle point if both [latex]f_{x}(x_{0},y_{0})=0[\/latex] and [latex]f_{y}(x_{0},y_{0})=0[\/latex], but [latex]f[\/latex]\u00a0does not have a local extremum at\u00a0[latex](x_{0},y_{0})[\/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-172\">\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 3. <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\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/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-3\/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":33,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/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-172","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/172","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/172\/revisions"}],"predecessor-version":[{"id":3769,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/172\/revisions\/3769"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/172\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=172"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=172"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=172"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=172"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}