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I have (probably) a fundamental problem understanding something related critical points and Lagrange multipliers.

As we know, if a function assumes an extreme value in an interior point of some open set, then the gradient of the function is 0.

Now, when dealing with constraint optimization using Lagrange multipliers, we also find an extreme value of the function restricted to some curve.

So why in the case of constraint optimization can't we also search for points where the gradient is 0? What am I missing here?

Thank you.

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# I Lagrange multipliers and critical points

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