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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.) –  Jonas Meyer Nov 5 '12 at 17:03
Thank you for the clarification :). –  dreamer Nov 5 '12 at 18:19

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Even though the level curves are locally parallel, one set of curves is more widely spaced than the other. In other words, even though both functions increase the fastest in the same direction, one function still increases faster than the other so their gradients have the same direction but different lengths.

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