R is the reals. D is the closed unit disk x2 + y2 = 1 in R2. The function f : D → R has partial derivatives f1(x, y) and f2(x, y) and all f(x, y) ∈ [-1, 1]. Show that there is a point (a, b) in the interior of D such that f1(a, b)2 + f2(a, b)2 ≤ 16.
Solution
Consider f(x, y) + 2x2 + 2y2. 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 f1(a, b) = -4a, f2(a, b) = -4b and f12 + f22 = 16(a2 + b2) < 16.
© John Scholes
jscholes@kalva.demon.co.uk
14 Jan 2002