I have been given the following exercise as homework: "Find the critical points of the function $$f(x,y)=(x^2+y^2)\exp(y^2-x^2)$$ and determine whether they are maxima, minima or saddle points."
So, determining the three critical points $(-1,0), (0,0), (1,0)$ was pretty easily done. Determining that $(-1,0)$ and $(1,0)$ are saddle points and $(0,0)$ is a local minimum is more tedious.
My question goes: "Is there any easier way to check what kind of critical points these are than to brute force the whole way through?"
Perhaps just the differentiation can be done more easily with some trick?
Edit: I used the formula $D = f_{xx} \cdot f_{yy}-(f_{xy})^2$ to determine what kind of critical points they were.