Problem Solving Set 3

06 July 2012

1. Let f : ℝ^{2} → ℝ^{2} be given by f(x, y) = (x^{2} − y^{2})e^{−x^{2}−y^{2}} (a) Prove that f attains its minimum and maximum.

(b) Determine all points (x, y) such that

\frac{∂f}{∂x} = \frac{∂f}{∂y} = 0

and determine for which of them f has global or local minimum or maximum.

2. A sequence u(n) satisfies the relation

u(n + 2) = [u(n + 1)]^{2} − u(n),

with u(1) = 13 and u(2) = 45. Prove that 2012 divides infinitely many terms of the sequence.