Explain, with justification, the significance of the eigenvalues of the Hessian in classifying the critical points of a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. In what circumstances are the eigenvalues inconclusive in establishing the character of a critical point?

Consider the function on $\mathbb{R}^{2}$,

$$f(x, y)=x y e^{-\alpha\left(x^{2}+y^{2}\right)}$$

Find and classify all of its critical points, for all real $\alpha$. How do the locations of the critical points change as $\alpha \rightarrow 0$ ?