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IMC 1994 D2 P2

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March 6, 2017
IMCreal analysis

Problem Statement

Let f ⁣:R2→Rf\colon \mathbb R ^2 \rightarrow \mathbb R be given by f(x,y)=(x2−y2)e−x2−y2f(x,y)=(x^2-y^2)e^{-x^2-y^2}.
a) Prove that ff attains its minimum and its maximum.
b) Determine all points (x,y)(x,y) such that ∂f∂x(x,y)=∂f∂y(x,y)=0\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0 and determine for which of them ff has global or local minimum or maximum.
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