(a,b) is stationary $\displaystyle \Leftrightarrow gradient(a,b)=0$

So, both $\displaystyle f_x=\frac{\partial f}{\partial x}=2xye^{x^2-y^2}$ and $\displaystyle f_y=\frac{\partial f}{\partial y}=(1-2y^2)e^{x^2-y^2}$ must equal $\displaystyle 0$.

Thus, two stationary points are $\displaystyle T_1=(0,\frac{1}{\sqrt2})$ and $\displaystyle T_2=(0,-\frac{1}{\sqrt2})$.

Now, how do I find if there are any "local extremes" among these two points, and what are they (MAX, min)?

I know you need

Hesse's matrix but don't know how.