The question is to find all the critical points of the function, then determine whether each point is either a minimum, maximum or saddle point.
The function is: $f(x,y) = x^3 - x^2 - x + y^3 - 2y^2$
I found that the critical points are:
$$P_1 = (1, 0),\quad P_2 = (-1/3, 0),\quad P_3 = (1, 4/3),\quad P_4 = (-1/3, 4/3)$$
I found the hessian matrix to be (for any $P$): $$H = \begin{pmatrix}6x-2 & 0\\ 0 & -4+6y \end{pmatrix} $$
The problem is when I want to determine for $P_1$ and $P_2$, since their determinants equal to $0$.
I don't know how to solve for these two cases.