Exercise 22

Show that \(f(x, y) = x^2ye^{-x^2-y^2}\) 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?