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| − | A '''Lagrangian multiplier''' is a constant, usually represented by the Greek symbol lambda (λ), which is part of the formula for finding an extreme value of a function ''f(x,y)'' subject to a constraint ''g(x,y)''. For example, the method of Lagrange multipliers can help determine the maximum area of a rectangle that can be fit inside a given circle represented by ''g(x,y) = x<sup>2</sup> + y<sup>2</sup>''. | + | A '''Lagrange multiplier''' is a constant, usually represented by the Greek symbol lambda (λ), which is part of the formula for finding an extreme value of a function ''f(x,y)'' subject to a constraint ''g(x,y)''. For example, the method of Lagrange multipliers can help determine the maximum area of a rectangle that can be fit inside a given circle represented by ''g(x,y) = x<sup>2</sup> + y<sup>2</sup>''. |
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| | The points at which [[extrema]] occur for a function ''f(x,y)'' subject to the constraint ''g(x,y)'' are the common solutions to: | | The points at which [[extrema]] occur for a function ''f(x,y)'' subject to the constraint ''g(x,y)'' are the common solutions to: |
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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? | | ''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: | + | ''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 Lagrange multiplier above, and setting the ''x'' and ''y'' coefficients equal to each other, we can generate these two equations: |
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| | :4y = λ2x | | :4y = λ2x |