Difference between revisions of "Math 22 Lagrange Multipliers"

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==Method of Lagrange Multipliers==
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  If <math>f(x,y)</math> has a maximum or minimum subject to the constraint <math>g(x,y)=0</math>, then it will occur at one of the critical numbers of the function  defined by
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  <math>F(x,y,\lambda)=f(x,y)-\lambda g(x,y)</math>.
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  In this section, we need to set up the system of equations:
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  <math>F_x(x,y,\lambda)=0</math>
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  <math>F_y(x,y,\lambda)=0</math>
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  <math>F_{\lambda}(x,y,\lambda)=0</math>
  
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'''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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|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>
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|-
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|<math>F_x(x,y,\lambda)=y-\lambda</math>
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|-
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|<math>F_y(x,y,\lambda)=x-\lambda</math>
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|-
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|<math>F_{\lambda}(x,y,\lambda)=-x-y+14</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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|So, <math>F(x,y,\lambda)=f(x,y)-\lambda g(x,y)=xy-\lambda (x+3y-6)=xy-\lambda x -3\lambda y+6\lambda</math>
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|-
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|<math>F_x(x,y,\lambda)=y-\lambda</math>
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|-
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|<math>F_y(x,y,\lambda)=x-3\lambda</math>
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|-
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|<math>F_{\lambda}(x,y,\lambda)=-x-3y+6</math>
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|}
  
  

Latest revision as of 08:57, 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
 .
 
 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:  
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+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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