R is the reals. f : R → R has a continuous derivative, f(0) = 0, and |f '(x)| <= |f(x)| for all x. Show that f is constant.
Suppose f is not constant. Then take an interval [a, b] of length < 1/2 such that f(a) = 0, f(b) ≠ 0, and |f(b)| ≥ |f(x)| for x ∈ [a, b]. Now applying the mean value theorem to the interval gives an immediate contradiction.
6th Putnam 1946
© John Scholes
14 Dec 1999