Let $\alpha \in \mathbb{R}$ and $f:\mathbb{R}^2\mapsto \mathbb{R}, \ f(x,y)=e^{-x}+\alpha x^2 - y^2$. Does $f$ have a local minimum?
I want to prove this with derivative test but since there is $e^{-x}$, $f$ is always differentialble and all the derivatives can never be zero for second, third derivative and so on, but except the first derivative $\begin{bmatrix}-e^{-x}+2\alpha x , -2y \end{bmatrix}$, when $y=0$ and $2\alpha x= e^{-x}$.
Is there any other way that can be used to prove a function that has a local extrema?