Q: Find the local maximum and minimum values and saddle point(s) of the function

$\displaystyle f(x, y) = (x^2 + y^2)e^{y^2 - x^2}$

I've found the the function's gradiant vector:

$\displaystyle \triangledown{f} = [-2xe^{y^2-x^2}(x^2+y^2-1), 2ye^{y^2-x^2}(x^2+y^2+1)]$

but I'm having trouble manipulating these equations to find where $\displaystyle \triangledown{f}=\vec{0}$. BOB says these points are $\displaystyle (0, 0)$ and $\displaystyle (\pm{1}, 0)$, but I'm not sure how to go about proving these are in fact the only points.