Using minors to determine type of extrema

Question

I've found this page for calculating the minors (Hauptminoren) of the Hessian matrix to determine which of the critical points of the matrix correspond to a maximum, a minimum or a saddle point. They say the matrix is positive definite if all $|q_A| > 0$ and negative if $|q_A|(-1)^k>0$, and a saddle point if otherwise. But what happens if one of $|q_A|$ is zero? No conclusion possible?

Answer 1

Classical example is the function

$$F: \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto x^4+y^4$$

which has a minimum at $(0,0)$ but the Hessian matrix is the zero matrix in that point.

On the other hand,

$$F: \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto x^3+y^3$$

also has Hessian matrix equal to the zero matrix in $(0,0)$, but this time you don't have a minimum or maximum.

So, the Hessian alone doesn't allow you to conclude in these cases.