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Lagrange multiplier (view source)

Revision as of 23:52, December 25, 2009

894 bytes added ,  23:52, December 25, 2009
example worked out
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and ''g(x,y) = 0''.
 
and ''g(x,y) = 0''.
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== Example ==
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''Problem'': What is the shape and dimension of the largest rectangle that fits within a circle having a radius equal to the square root of 2?
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''Solution'':  ''g(x,y) = x<sup>2</sup> + y<sup>2</sup> - 2 = 0''.  We want to maximize the area of a rectangle.  Assuming symmetry about the origin (we could always map a solution to a symmetric one), the area ''f(x,y)'' that we want to maximize equals 2x times 2y:  ''f(x,y) = 4xy''.  From the equation for the Lagrangian multiplier above, and setting the ''x'' and ''y'' coefficients equal to each other, we can generate these two equations:
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:4y = \lambda 2x
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:4x = \lambda 2y
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Solving each for lambda (λ) demonstrates that x must be equal to y, and plugging that into the equation for ''g'' yields:
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:2x<sup>2</sup> = 2
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:x = 1 = y
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Then do not forget to include the maximum area in your answer, which is ''f(1,1) = 4(1)(1) = 4.
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[[Category:calculus]]
 
[[Category:calculus]]
 
[[Category:mathematics]]
 
[[Category:mathematics]]
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