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## Homework Statement

Suppose that the function f: R

^{n}--> R has first-order partial derivatives and that the point

**x**in R

^{n}is a local minimizer for f: R

^{n}--> 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**.

## Homework Equations

Df(

**x**)=(df/dx

_{1},df/dx

_{2},...,df/dx

_{n})

## 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?