[Math] Classifying the stationary points of $f(x, y) = 4xy-x^4-y^4 $

Question

$f(x, y) = 4xy-x^4-y^4 $

The gradient of this function is $0$ in $(-1, -1), (0, 0),(1, 1)$

I tried to compute the determinant of the Hessian Matrix, but it's 0 for every point, I always get a null eigenvalue for each point, so no matter what I can't find out what kind of points these are.

For $(0, 0)$, since $f(0, 0) = 0$, $f(x, 0) = -x^4$ and $f(0, y) = -y^4$. Since both will always be negative, isn't that supposed to be a maximum or a minimum point? According to Wolfram it's a saddle point and I can't understand why.

Answer 1

Note that:

• $f(x,x)=4x^2-2x^4=2x^2(2-x^2)$ and therefore $f(x,x)>0$ of $x\in\left(-\sqrt 2,\sqrt2\right)\setminus\{0\}$;
• $f(x,0)=-x^4<0$ (unless $x=0$).

Therefore, yes, $(0,0)$ is a saddle point.