Chapter 14.7, Problem 22E

Show that $f\left(x,y\right)=x^{2}y{e}^{-x^2-y^2}$ has maximum values at $(\pm 1,1/\sqrt{2})$ and minimum values at $(\pm 1,-1/\sqrt{2})$. Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give rise to maximum values? Minimum values? Saddle points?