{"id":648,"date":"2025-10-09T15:28:03","date_gmt":"2025-10-09T15:28:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=648"},"modified":"2025-10-15T15:11:02","modified_gmt":"2025-10-15T15:11:02","slug":"optimization-and-applications-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/optimization-and-applications-cheat-sheet\/","title":{"raw":"Optimization and Applications: Cheat Sheet","rendered":"Optimization and Applications: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Critical Points and Local Extrema<\/strong>\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<strong>Constrained Optimization<\/strong>\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<h2>Key Equations<\/h2>\r\n<ul>\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 \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>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>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>\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":"<h2>Essential Concepts<\/h2>\n<p><strong>Critical Points and Local Extrema<\/strong><\/p>\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<p><strong>Constrained Optimization<\/strong><\/p>\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<h2>Key Equations<\/h2>\n<ul>\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<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>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>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<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","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-648","chapter","type-chapter","status-publish","hentry"],"part":430,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/648","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/648\/revisions"}],"predecessor-version":[{"id":1188,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/648\/revisions\/1188"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/430"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/648\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=648"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=648"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=648"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=648"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}