Putnam 1994 Problem B3 Let R be the reals and R + the positive reals. f : R → R + is differentiable and f '(x) > f(x) for all x. For what k must f(x) exceed e kx for all sufficiently large k? Solution
: k ≤ 1. Let g(x) = e -x f(x), then g'(x) = e -x (f '(x) - f(x) ) > 0 for all x, so g is a strictly increasing function. Hence for any h  e hx for all sufficiently large x. Hence if k  e kx for all sufficiently large x. The tricky case is obviously k = 1. The argument above suggests that if g(x) keeps below 1, then we will have f(x)  f(x), but f(x) < e kx for all x, provided k ≥ 1. Putnam 1994 © John Scholes jscholes@kalva.demon.co.uk 12 Dec 1998