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.
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?