Difference between revisions of "Math 22 Lagrange Multipliers"
| Line 8: | Line 8: | ||
<math>F_y(x,y,\lambda)=0</math> | <math>F_y(x,y,\lambda)=0</math> | ||
<math>F_{\lambda}(x,y,\lambda)=0</math> | <math>F_{\lambda}(x,y,\lambda)=0</math> | ||
| − | + | ||
| − | + | '''Example:''' Set up the Lagrange Multipliers: | |
| + | |||
| + | '''1)''' Maximum: <math>f(x,y)=xy</math> and Constraint <math>x+y-14=0</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>\frac{\partial z}{\partial x}=4x^2-4y</math> | ||
| + | |- | ||
| + | |<math>\frac{\partial z}{\partial y}=-4x</math> | ||
| + | |} | ||
| + | |||
| + | '''2)''' Maximum: <math>f(x,y)=xy</math> and Constraint <math>x+3y-6=0</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>\frac{\partial z}{\partial x}=2xy^3</math> | ||
| + | |- | ||
| + | |<math>\frac{\partial z}{\partial y}=3x^2y^2</math> | ||
| + | |} | ||
Revision as of 08:50, 18 August 2020
Method of Lagrange Multipliers
If has a maximum or minimum subject to the constraint , then it will occur at one of the critical numbers of the function defined by . In this section, we need to set up the system of equations: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F_{x}(x,y,\lambda )=0} Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F_{y}(x,y,\lambda )=0} Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F_{\lambda }(x,y,\lambda )=0}
Example: Set up the Lagrange Multipliers:
1) Maximum: and Constraint Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+y-14=0}
| Solution: |
|---|
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial x}=4x^2-4y} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial y}=-4x} |
2) Maximum: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y)=xy} and Constraint Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+3y-6=0}
| Solution: |
|---|
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial x}=2xy^3} |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial z}{\partial y}=3x^2y^2} |
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