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?
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 $0$.