Math 22 Lagrange Multipliers
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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: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x,y)=xy} 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: |
|---|
| So, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x,y,\lambda )=f(x,y)-\lambda g(x,y)=xy-\lambda (x+y-14)=xy-\lambda x-\lambda y+14\lambda } |
| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F_{x}(x,y,\lambda )=y-\lambda } |
| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F_{y}(x,y,\lambda )=x-\lambda } |
| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F_{\lambda }(x,y,\lambda )=-x-y+14} |
2) Maximum: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(x,y)=xy} and Constraint Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+3y-6=0}
| Solution: |
|---|
| So, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x,y,\lambda )=f(x,y)-\lambda g(x,y)=xy-\lambda (x+3y-6)=xy-\lambda x-3\lambda y+6\lambda } |
| 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(x,y,\lambda)=y-\lambda} |
| 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_y(x,y,\lambda)=x-3\lambda} |
| 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_{\lambda}(x,y,\lambda)=-x-3y+6} |
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