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 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,\lambda)=f(x,y)-\lambda g(x,y)} . In this section, we need to set up the system of equations: 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)=0} 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)=0} 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)=0}
Example: Set up the Lagrange Multipliers:
1) 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+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} |
This page were made by Tri Phan