1. The problem statement, all variables and given/known data Suppose that the function f: Rn --> R has first-order partial derivatives and that the point x in Rn is a local minimizer for f: Rn --> R, meaning that there is a positive number r such that f(x+h) > f(x) if dist(x,x+h) < r. Prove that Df(x)=0. 2. Relevant equations Df(x)=(df/dx1,df/dx2,...,df/dxn) 3. The attempt at a solution We know that the function has first-order partial derivatives, which makes finding the gradient vector possible. And the definition for local minimizer is already given in the problem. I just need to prove that all partial derivatives are equal to zero. But how does knowing the local minimizer help me figure out the gradient vector?