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I do not fully understand the 'definition' of the Lagrange multipliers. I do understand that a maximum occurs when the constraint and the objective function are tangent to eachother. However, I do not understand why this implies that the $\nabla{f}=\lambda\times\nabla{g}$. Why is it not true that the gradient of $f$ IS EQUAL to the gradient of $g$? Doesn't the fact that the level curves are parallel imply that the derivatives are equal and thus that the gradients are equal (and not a multiple of eachother)?

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    For example, optimizing under the constraint $x^2+y^2 = 1$ is the same as optimizing under the constraint $2x^2+2y^2 = 2$, but $x^2+y^2$ and $2x^2+2y^2$ have different gradients. The direction of the level curves determine the direction of the gradient, but not the magnitude. (It is the spacing that determines magnitude, as Nick answered.)2012-11-05
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    Thank you for the clarification :).2012-11-05

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