Difference between revisions of "Math 22 Lagrange Multipliers"

From Math Wiki
Jump to navigation Jump to search
Line 15: Line 15:
 
!Solution:  
 
!Solution:  
 
|-
 
|-
−
|<math>\frac{\partial z}{\partial x}=4x^2-4y</math>
+
|So, <math>F(x,y,\lambda)=f(x,y)-\lambda g(x,y)=xy-\lambda (x+y-14)=xy-\lambda x -\lambda y+14\lambda</math>
 
|-
 
|-
−
|<math>\frac{\partial z}{\partial y}=-4x</math>
+
|<math>F_x(x,y,\lambda)=y-\lambda</math>
 +
|-
 +
|<math>F_y(x,y,\lambda)=x-\lambda</math>
 +
|-
 +
|<math>F_{\lambda}(x,y,\lambda)=-x-y+14</math>  
 
|}
 
|}
  

Revision as of 08:54, 18 August 2020

Method of Lagrange Multipliers

 If 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)}
 has a maximum or minimum subject to the 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 g(x,y)=0}
, 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:  
So, 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)=xy-\lambda (x+y-14)=xy-\lambda x -\lambda y+14\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-\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-y+14}

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}


Return to Topics Page

This page were made by Tri Phan

Retrieved from "https://wiki.math.ucr.edu/index.php?title=Math_22_Lagrange_Multipliers&oldid=2608"