{"id":657,"date":"2025-10-09T15:28:26","date_gmt":"2025-10-09T15:28:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=657"},"modified":"2025-10-29T14:39:12","modified_gmt":"2025-10-29T14:39:12","slug":"constrained-optimization-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/constrained-optimization-fresh-take\/","title":{"raw":"Constrained Optimization: Fresh Take","rendered":"Constrained Optimization: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve optimization problems with one constraint<\/li>\r\n \t<li>Solve optimization problems with two constraints<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Lagrange Multipliers<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">When you need to optimize a function subject to a constraint, Lagrange multipliers provide an elegant alternative to parameterizing boundaries. The key insight: at an optimum point, the gradient of your objective function must be parallel to the gradient of your constraint.<\/p>\r\n<p class=\"whitespace-normal break-words\">To maximize or minimize [latex]f(x, y)[\/latex] subject to constraint [latex]g(x, y) = 0[\/latex], look for points where:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\nabla f(x_0, y_0) = \\lambda \\nabla g(x_0, y_0)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The scalar [latex]\\lambda[\/latex] (lambda) is called the Lagrange multiplier. Geometrically, this says the level curves of [latex]f[\/latex] and [latex]g[\/latex] are tangent at the optimal point.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Identify your functions<\/strong>: Determine the objective function [latex]f[\/latex] (what you're optimizing) and constraint [latex]g[\/latex] (rewritten so it equals zero)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Set up the system<\/strong>: Write [latex]\\nabla f = \\lambda \\nabla g[\/latex] and [latex]g = 0[\/latex], which gives you three equations in three unknowns<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Solve the system<\/strong>: Find [latex]x_0[\/latex], [latex]y_0[\/latex], and [latex]\\lambda[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Evaluate and compare<\/strong>: Calculate [latex]f[\/latex] at each solution point. Test a few other constraint-satisfying points to verify you found a maximum or minimum<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Why This Works<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">If [latex]f[\/latex] has an extremum on the constraint curve [latex]g = 0[\/latex], then moving along the constraint shouldn't change [latex]f[\/latex] to first order. This happens when [latex]\\nabla f[\/latex] is perpendicular to the constraint curve\u2014which means [latex]\\nabla f[\/latex] must be parallel to [latex]\\nabla g[\/latex] (since [latex]\\nabla g[\/latex] is also perpendicular to the curve).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Extension to Three Variables<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]f(x, y, z)[\/latex] with constraint [latex]g(x, y, z) = 0[\/latex], the method is identical: solve [latex]\\nabla f = \\lambda \\nabla g[\/latex] and [latex]g = 0[\/latex], giving four equations in four unknowns.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Use the method of Lagrange multipliers to find the maximum value of\u00a0[latex]f(x, y)=9x^{2}+36xy-4y^{2}-18x-8y[\/latex] subject to the constraint\u00a0[latex]3x+4y=32[\/latex].[reveal-answer q=\"881245602\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"881245602\"]\r\n\r\n[latex]f[\/latex] has a maximum value of [latex]976[\/latex] at the point\u00a0[latex](8, 2)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A company has determined that its production level is given by the Cobb-Douglas function [latex]f(x,y)=2.5x^{0.45}y^{0.55}[\/latex] where [latex]x[\/latex] represents the total number of labor hours in [latex]1[\/latex] year and [latex]y[\/latex] represents the total capital input for the company. Suppose\u00a0[latex]1[\/latex] unit of labor costs [latex]$40[\/latex] and\u00a0[latex]1[\/latex] unit of capital costs [latex]$50[\/latex]. Use the method of Lagrange multipliers to find the maximum value of [latex]f(x,y)=2.5x^{0.45}y^{0.55}[\/latex] subject to a budgetary constraint of [latex]$500,000[\/latex] per year.[reveal-answer q=\"625684923\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"625684923\"]\r\n\r\nA maximum production level of [latex]13,890[\/latex] occurs with [latex]5,625[\/latex] labor hours and [latex]$5,500[\/latex] of total capital input.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186170&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=xu2_ZpyYkSg&amp;video_target=tpm-plugin-ky4vc3gh-xu2_ZpyYkSg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.38_transcript.html\">transcript for \u201c4.38\u201d here (opens in new window).<\/a><\/center><\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1167794177145\">Use the method of Lagrange multipliers to find the minimum value of the function<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{f(x,y,z)=x+y+z}[\/latex]<\/p>\r\n<span style=\"font-size: 1rem; text-align: initial;\">subject to the constraint [latex]x^2+y^2+z^2=1[\/latex].<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0<\/span>\r\n\r\n[reveal-answer q=\"457389340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"457389340\"]\r\n\r\n[latex]\\hspace{6cm}\\begin{align}\r\n\r\nf\\left(\\frac{\\sqrt3}3,\\frac{\\sqrt3}3,\\frac{\\sqrt3}3\\right)&amp;=\\frac{\\sqrt3}3+\\frac{\\sqrt3}3+\\frac{\\sqrt3}3=\\sqrt3 \\\\\r\n\r\nf\\left(-\\frac{\\sqrt3}3,-\\frac{\\sqrt3}3,-\\frac{\\sqrt3}3\\right)&amp;=-\\frac{\\sqrt3}3-\\frac{\\sqrt3}3-\\frac{\\sqrt3}3=-\\sqrt3\r\n\r\n\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Problems with Two Constraints<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">When optimizing a function of three variables subject to two constraints, you need two Lagrange multipliers. The method extends naturally from the single-constraint case, but the algebra becomes more involved.<\/p>\r\n<p class=\"whitespace-normal break-words\">To optimize [latex]f(x, y, z)[\/latex] subject to two constraints [latex]g(x, y, z) = 0[\/latex] and [latex]h(x, y, z) = 0[\/latex], solve:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\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)[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">along with [latex]g(x_0, y_0, z_0) = 0[\/latex] and [latex]h(x_0, y_0, z_0) = 0[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>What This Means<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">At an optimal point, the gradient of [latex]f[\/latex] must lie in the plane spanned by the gradients of [latex]g[\/latex] and [latex]h[\/latex]. Geometrically, you're finding extrema along the curve where two surfaces intersect.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Identify your functions<\/strong>: Write down [latex]f[\/latex] (objective), [latex]g[\/latex] (first constraint), and [latex]h[\/latex] (second constraint), each set equal to zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Calculate all three gradients<\/strong>: [latex]\\nabla f[\/latex], [latex]\\nabla g[\/latex], and [latex]\\nabla h[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Set up the system<\/strong>: Write [latex]\\nabla f = \\lambda_1 \\nabla g + \\lambda_2 \\nabla h[\/latex], [latex]g = 0[\/latex], and [latex]h = 0[\/latex]\u2014this gives five equations in five unknowns<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Solve systematically<\/strong>: Use substitution and elimination to reduce variables. Look for relationships between [latex]\\lambda_1[\/latex] and [latex]\\lambda_2[\/latex] that simplify the system<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Evaluate [latex]f[\/latex]<\/strong> at all solution points and compare to find maximum and minimum values<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\"><strong>Critical Tip<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The algebra gets complex quickly. Work methodically, solve equations in stages, and check your arithmetic carefully. Often you'll eliminate one multiplier, then the other, before solving for [latex]x_0[\/latex], [latex]y_0[\/latex], and [latex]z_0[\/latex].<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1167794314954\">Use the method of Lagrange multipliers to find the minimum value of the function<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{f(x,y,z)=x^2+y^2+z^2}[\/latex]<\/p>\r\n<span style=\"font-size: 1rem; text-align: initial;\">subject to the constraints [latex]2x+y+2z=9[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">and [latex]5x+5y+7z=29[\/latex].\u00a0<\/span>\r\n\r\n[reveal-answer q=\"131765238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"131765238\"]\r\n\r\n[latex]f(2,1,2)=9[\/latex] is a minimum.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186171&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=xwTxfOqbXkw&amp;video_target=tpm-plugin-jwom32th-xwTxfOqbXkw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.40_transcript.html\">transcript for \u201cCP 4.40\u201d here (opens in new window).<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve optimization problems with one constraint<\/li>\n<li>Solve optimization problems with two constraints<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Lagrange Multipliers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When you need to optimize a function subject to a constraint, Lagrange multipliers provide an elegant alternative to parameterizing boundaries. The key insight: at an optimum point, the gradient of your objective function must be parallel to the gradient of your constraint.<\/p>\n<p class=\"whitespace-normal break-words\">To maximize or minimize [latex]f(x, y)[\/latex] subject to constraint [latex]g(x, y) = 0[\/latex], look for points where:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\nabla f(x_0, y_0) = \\lambda \\nabla g(x_0, y_0)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The scalar [latex]\\lambda[\/latex] (lambda) is called the Lagrange multiplier. Geometrically, this says the level curves of [latex]f[\/latex] and [latex]g[\/latex] are tangent at the optimal point.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Identify your functions<\/strong>: Determine the objective function [latex]f[\/latex] (what you&#8217;re optimizing) and constraint [latex]g[\/latex] (rewritten so it equals zero)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Set up the system<\/strong>: Write [latex]\\nabla f = \\lambda \\nabla g[\/latex] and [latex]g = 0[\/latex], which gives you three equations in three unknowns<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Solve the system<\/strong>: Find [latex]x_0[\/latex], [latex]y_0[\/latex], and [latex]\\lambda[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Evaluate and compare<\/strong>: Calculate [latex]f[\/latex] at each solution point. Test a few other constraint-satisfying points to verify you found a maximum or minimum<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Why This Works<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">If [latex]f[\/latex] has an extremum on the constraint curve [latex]g = 0[\/latex], then moving along the constraint shouldn&#8217;t change [latex]f[\/latex] to first order. This happens when [latex]\\nabla f[\/latex] is perpendicular to the constraint curve\u2014which means [latex]\\nabla f[\/latex] must be parallel to [latex]\\nabla g[\/latex] (since [latex]\\nabla g[\/latex] is also perpendicular to the curve).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Extension to Three Variables<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For [latex]f(x, y, z)[\/latex] with constraint [latex]g(x, y, z) = 0[\/latex], the method is identical: solve [latex]\\nabla f = \\lambda \\nabla g[\/latex] and [latex]g = 0[\/latex], giving four equations in four unknowns.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Use the method of Lagrange multipliers to find the maximum value of\u00a0[latex]f(x, y)=9x^{2}+36xy-4y^{2}-18x-8y[\/latex] subject to the constraint\u00a0[latex]3x+4y=32[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q881245602\">Show Solution<\/button><\/p>\n<div id=\"q881245602\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f[\/latex] has a maximum value of [latex]976[\/latex] at the point\u00a0[latex](8, 2)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A company has determined that its production level is given by the Cobb-Douglas function [latex]f(x,y)=2.5x^{0.45}y^{0.55}[\/latex] where [latex]x[\/latex] represents the total number of labor hours in [latex]1[\/latex] year and [latex]y[\/latex] represents the total capital input for the company. Suppose\u00a0[latex]1[\/latex] unit of labor costs [latex]$40[\/latex] and\u00a0[latex]1[\/latex] unit of capital costs [latex]$50[\/latex]. Use the method of Lagrange multipliers to find the maximum value of [latex]f(x,y)=2.5x^{0.45}y^{0.55}[\/latex] subject to a budgetary constraint of [latex]$500,000[\/latex] per year.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q625684923\">Show Solution<\/button><\/p>\n<div id=\"q625684923\" class=\"hidden-answer\" style=\"display: none\">\n<p>A maximum production level of [latex]13,890[\/latex] occurs with [latex]5,625[\/latex] labor hours and [latex]$5,500[\/latex] of total capital input.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186170&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=xu2_ZpyYkSg&amp;video_target=tpm-plugin-ky4vc3gh-xu2_ZpyYkSg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.38_transcript.html\">transcript for \u201c4.38\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1167794177145\">Use the method of Lagrange multipliers to find the minimum value of the function<\/p>\n<p style=\"text-align: center;\">[latex]\\large{f(x,y,z)=x+y+z}[\/latex]<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">subject to the constraint [latex]x^2+y^2+z^2=1[\/latex].<\/span><span style=\"font-size: 1rem; text-align: initial;\">\u00a0<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q457389340\">Show Solution<\/button><\/p>\n<div id=\"q457389340\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\hspace{6cm}\\begin{align}    f\\left(\\frac{\\sqrt3}3,\\frac{\\sqrt3}3,\\frac{\\sqrt3}3\\right)&=\\frac{\\sqrt3}3+\\frac{\\sqrt3}3+\\frac{\\sqrt3}3=\\sqrt3 \\\\    f\\left(-\\frac{\\sqrt3}3,-\\frac{\\sqrt3}3,-\\frac{\\sqrt3}3\\right)&=-\\frac{\\sqrt3}3-\\frac{\\sqrt3}3-\\frac{\\sqrt3}3=-\\sqrt3    \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Problems with Two Constraints<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When optimizing a function of three variables subject to two constraints, you need two Lagrange multipliers. The method extends naturally from the single-constraint case, but the algebra becomes more involved.<\/p>\n<p class=\"whitespace-normal break-words\">To optimize [latex]f(x, y, z)[\/latex] subject to two constraints [latex]g(x, y, z) = 0[\/latex] and [latex]h(x, y, z) = 0[\/latex], solve:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\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)[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">along with [latex]g(x_0, y_0, z_0) = 0[\/latex] and [latex]h(x_0, y_0, z_0) = 0[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>What This Means<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">At an optimal point, the gradient of [latex]f[\/latex] must lie in the plane spanned by the gradients of [latex]g[\/latex] and [latex]h[\/latex]. Geometrically, you&#8217;re finding extrema along the curve where two surfaces intersect.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Identify your functions<\/strong>: Write down [latex]f[\/latex] (objective), [latex]g[\/latex] (first constraint), and [latex]h[\/latex] (second constraint), each set equal to zero<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Calculate all three gradients<\/strong>: [latex]\\nabla f[\/latex], [latex]\\nabla g[\/latex], and [latex]\\nabla h[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Set up the system<\/strong>: Write [latex]\\nabla f = \\lambda_1 \\nabla g + \\lambda_2 \\nabla h[\/latex], [latex]g = 0[\/latex], and [latex]h = 0[\/latex]\u2014this gives five equations in five unknowns<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Solve systematically<\/strong>: Use substitution and elimination to reduce variables. Look for relationships between [latex]\\lambda_1[\/latex] and [latex]\\lambda_2[\/latex] that simplify the system<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Evaluate [latex]f[\/latex]<\/strong> at all solution points and compare to find maximum and minimum values<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\"><strong>Critical Tip<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The algebra gets complex quickly. Work methodically, solve equations in stages, and check your arithmetic carefully. Often you&#8217;ll eliminate one multiplier, then the other, before solving for [latex]x_0[\/latex], [latex]y_0[\/latex], and [latex]z_0[\/latex].<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1167794314954\">Use the method of Lagrange multipliers to find the minimum value of the function<\/p>\n<p style=\"text-align: center;\">[latex]\\large{f(x,y,z)=x^2+y^2+z^2}[\/latex]<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">subject to the constraints [latex]2x+y+2z=9[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">and [latex]5x+5y+7z=29[\/latex].\u00a0<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q131765238\">Show Solution<\/button><\/p>\n<div id=\"q131765238\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(2,1,2)=9[\/latex] is a minimum.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186171&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=xwTxfOqbXkw&amp;video_target=tpm-plugin-jwom32th-xwTxfOqbXkw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.40_transcript.html\">transcript for \u201cCP 4.40\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":12,"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-657","chapter","type-chapter","status-publish","hentry"],"part":430,"module-header":"fresh_take","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\/657","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":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/657\/revisions"}],"predecessor-version":[{"id":1840,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/657\/revisions\/1840"}],"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\/657\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=657"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=657"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=657"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=657"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}