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Given the function: $f(x, y) = x^2 - y^2$

Determine the local and global extrema for f in the set $M = \{(x, y) \in \mathbb{R^2}\ |\ y + e^{-x^2} -1 =0\}$

I know about Lagrange Multipliers, but we didn't cover them yet, so I'm wondering if there's any way to do this without using them?

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No, you don't need Lagrange multipliers here. Just define$$\varphi(x)=x^2-\left(1-e^{-x^2}\right)^2.$$This is a differentiable function from $\mathbb R$ into itself. Use the standard methods for finding its local extrema. You will get that it has no local maximum and that its only local minimum is located at $0$.

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