R is the reals. D is the closed unit disk x^{2} + y^{2} = 1 in R^{2}. The function f **:** D → R has partial derivatives f_{1}(x, y) and f_{2}(x, y) and all f(x, y) ∈ [-1, 1]. Show that there is a point (a, b) in the interior of D such that f_{1}(a, b)^{2} + f_{2}(a, b)^{2} ≤ 16.

**Solution**

Consider f(x, y) + 2x^{2} + 2y^{2}. It is at least 1 on the entire boundary of D and at most 1 at the centre. So it is either constant at an interior point of D or has a minimum at an interior point. In either case, there is an interior point (a, b) at which its two partial derivatives are zero. So f_{1}(a, b) = -4a, f_{2}(a, b) = -4b and f_{1}^{2} + f_{2}^{2} = 16(a^{2} + b^{2}) < 16.

© John Scholes

jscholes@kalva.demon.co.uk

14 Jan 2002