1. The problem statement, all variables and given/known data Suppose that f is a differentiable real function in an open set E (which is a subset of) ℝn, and that that f has a local maximum at a point x in E. Prove that f'(x)=0. 2. Relevant equations Definition. Suppose E is an open set in ℝn, f maps E into ℝm, and x is an element of E. If there exists a linear transformation A of ℝn into ℝm such that limh→0 |f(x+h)-f(x)-Ah|/|h|=0, then f is differentiable at x, and we write f'(x)=A. 3. The attempt at a solution f has a local max at x, so there exists a δ>0 such that f(y)≤f(y) for all y at which |y-x|<δ. Let h=x+y. Whenever |h|≤δ we have f(x)≥f(x+h) or f(x)≥f(x-h). Now I need to use some algebraic cut-and-paste wizardry to end up with |f(x+h)-f(x)|/|h|. Could I please get an inconspicuous suggestion from the peanut gallery here?