Difference between revisions of "Math 22 Lagrange Multipliers"

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   <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:
 +
 
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'''1)''' Maximum: <math>f(x,y)=xy</math> and Constraint <math>x+y-14=0</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>\frac{\partial z}{\partial x}=4x^2-4y</math>
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|-
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|<math>\frac{\partial z}{\partial y}=-4x</math>
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|}
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'''2)''' Maximum: <math>f(x,y)=xy</math> and Constraint <math>x+3y-6=0</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>\frac{\partial z}{\partial x}=2xy^3</math>
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|-
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|<math>\frac{\partial z}{\partial y}=3x^2y^2</math>
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|}
  
  

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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