{"id":174,"date":"2021-07-30T17:24:51","date_gmt":"2021-07-30T17:24:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=174"},"modified":"2022-10-29T02:18:27","modified_gmt":"2022-10-29T02:18:27","slug":"summary-of-lagrange-multipliers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-lagrange-multipliers\/","title":{"raw":"Summary of Lagrange Multipliers","rendered":"Summary of Lagrange Multipliers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>An objective function combined with one or more constraints is an example of an optimization problem.<\/li>\r\n \t<li>To solve optimization problems, we apply the method of Lagrange multipliers using a four-step problem-solving strategy.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Method of Lagrange multipliers, one constraint\r\n<\/strong>[latex]\\begin{array}{cc} \\nabla f(x_{0},y_{0}) &amp;= \\lambda\\nabla g(x_{0},y_{0}) \\\\ \\hfill g(x_{0},y_{0}) &amp;= 0 \\hfill \\end{array}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Method of Lagrange multipliers, two constraints<\/strong>\r\n[latex]\\begin{array}{cc} \\nabla f(x_{0},y_{0},z_{0}) &amp;= \\lambda_{1}\\nabla g(x_{0},y_{0},z_{0})+\\lambda_{2}\\nabla h(x_{0},y_{0},z_{0}) \\\\ \\hfill g(x_{0},y_{0},z_{0}) &amp;= 0 \\hfill\\\\ \\hfill h(x_{0},y_{0},z_{0}) &amp;= 0 \\hfill \\end{array}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>constraint<\/dt>\r\n \t<dd>an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Lagrange Multiplier<\/dt>\r\n \t<dd>the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable\u00a0[latex]\\lambda[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>method of Lagrange multipliers<\/dt>\r\n \t<dd>a method of solving an optimization problem subject to one or more constraints<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>objective function<\/dt>\r\n \t<dd>the function that is to be maximized or minimized in an optimization problem<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>optimization problem<\/dt>\r\n \t<dd>calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>An objective function combined with one or more constraints is an example of an optimization problem.<\/li>\n<li>To solve optimization problems, we apply the method of Lagrange multipliers using a four-step problem-solving strategy.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Method of Lagrange multipliers, one constraint<br \/>\n<\/strong>[latex]\\begin{array}{cc} \\nabla f(x_{0},y_{0}) &= \\lambda\\nabla g(x_{0},y_{0}) \\\\ \\hfill g(x_{0},y_{0}) &= 0 \\hfill \\end{array}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Method of Lagrange multipliers, two constraints<\/strong><br \/>\n[latex]\\begin{array}{cc} \\nabla f(x_{0},y_{0},z_{0}) &= \\lambda_{1}\\nabla g(x_{0},y_{0},z_{0})+\\lambda_{2}\\nabla h(x_{0},y_{0},z_{0}) \\\\ \\hfill g(x_{0},y_{0},z_{0}) &= 0 \\hfill\\\\ \\hfill h(x_{0},y_{0},z_{0}) &= 0 \\hfill \\end{array}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>constraint<\/dt>\n<dd>an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Lagrange Multiplier<\/dt>\n<dd>the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable\u00a0[latex]\\lambda[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>method of Lagrange multipliers<\/dt>\n<dd>a method of solving an optimization problem subject to one or more constraints<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>objective function<\/dt>\n<dd>the function that is to be maximized or minimized in an optimization problem<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>optimization problem<\/dt>\n<dd>calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers<\/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-174\">\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":36,"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-174","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/174","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\/174\/revisions"}],"predecessor-version":[{"id":3770,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/174\/revisions\/3770"}],"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\/174\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=174"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=174"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=174"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=174"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}