The critical points and saddle points of the function $$f\left(x,y\right)=x^{3}+y^{2}-3xy$$ are $A=\left(0,0\right)$ and $B=\left(\frac{3}{2},\frac{9}{4}\right)$. The hessian is $$ H=\begin{bmatrix}\begin{array}{rr} 6x & -3\\ -3 & 2 \end{array}\end{bmatrix}. $$
$A$: Principal minors are $0$ and $-9$, so $A$ is a saddle point.
$B$: Principal minors are $9$ and $9$, so $B$ is a local minimum.
I would like to check the result from Wolfram Alpha, $B$ was okay but it gave $A$ as a min (or max) as in the pictures.
I also add the plot of the function which shows a saddle point as well.
Is my solution wrong?
It is beautifully seen at the picture that $A(0,0)$ is a saddle point. Something like mountain pass. Can you see this?