Explain, with justification, how the nature of a critical (stationary) point of a function $f(\mathbf{x})$ can be determined by consideration of the eigenvalues of the Hessian matrix $H$ of $f(\mathbf{x})$ if $H$ is non-singular. What happens if $H$ is singular?

Let $f(x, y)=\left(y-x^{2}\right)\left(y-2 x^{2}\right)+\alpha x^{2}$. Find the critical points of $f$ and determine their nature in the different cases that arise according to the values of the parameter $\alpha \in \mathbb{R}$.