Fermat's Theorem

Fermat’s Theorem: Supposse a function which it's defined on the interval (a, b), in which we can find c. 

If f(c) is an extreme value of f, and f is differentiable at c, then f’(c) = 0.

A maximum or minimum can occur where the derivative doesn't exist. And also, a point where the derivative is zero may not be a maximum or minimum point. For example, a function that increases, has a region with slope equals to 0 and continues increasing. It's not an extremum value.