Only one critical point and Hessian determinant is zero

Question

Suppose I have a function $f(x,y)$ and I want to determine the nature of its critical points.

Suppose only one critical point is found, but the Hessian matrix is not useful to determine the nature of the point (i.e. the Hessian determinant is zero in that point).

Is it true, that if

1. I calculate the value of the function in that point, and in arbitrary other points.
2. I find some values that are less than the function in the critical point, and some other values that are greater than the function in the critical point.

Then:

• The critical point must be a saddle point?

Please consider the assumptions, i.e. there is only one critical point and I found some values that are less and some that are greater.

EDIT: An example: consider the function $f(x,y)=x^4-y^4$.

There is only one critical point at $(0,0)$, but the Hessian determinant is zero.

Now, $f(0,0)=0$, and I can find other values like $f(1,0)=1$ and $f(0,1)=-1$.

Can I say with absolute certainty that $(0,0)$ is a saddle point?