Classifying the critical points of a multivariable function Given the function: $f(x,y)=\frac32x-\frac12x^3-xy^2$ Thanks to the gradient I managed to find that the critical points are: $(1,0),\ (-1,0),\ (0,-\sqrt{\frac32}),\ (0,+\sqrt\frac32)$ Then I found the Hessian matrix and calculated the discriminant. The problem is, when I try to find the "A" component of the discriminant (disc= AC-$B^2$) I get $0$ for the points (0,-$\sqrt\frac32$),(0,+$\sqrt\frac32$). although the discriminant itself is $>0$. So what are these points (0,-$\sqrt\frac32$),(0,+$\sqrt\frac32$) and how can I classify them?

For $q_\pm=(\pm 1,0)$, ${\rm det\ Hess}\ f>0$ : Hence it is local minimum or local maximum : $f(1,0)=1$ and $f(1+\epsilon,0)=1 \- \frac{3}{2}\epsilon^2 - \frac{1}{2}\epsilon^3$ so that $q_+$ is local maximum

Since $f(-1,0)=-1$ and $f(-1-\epsilon,0)=-f(1+\epsilon,0)$ so it is local minimum

For $p_\pm = (0,\pm\sqrt{\frac{3}{2}})$, $ {\rm det\ Hess}\ f <0$ Hence at $p_+$, there are two curves $\alpha_i$ s.t. $f\circ \alpha_1$ has local minimum and $f\circ \alpha_2$ has local maximum