IMC 1994 · Problem 2

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be given by $f(x,y) = (x^2 - y^2)e^{-x^2-y^2}$.

a) Prove that $f$ attains its minimum and its maximum.

b) Determine all points $(x,y)$ such that $\dfrac{\partial f}{\partial x}(x,y) = \dfrac{\partial f}{\partial y}(x,y) = 0$ and determine for which of them $f$ has global or local minimum or maximum.