If f is differentiable on the interval [a,b] and f'(a)<0<f'(b), prove that there is a c with a<c<b for which f'(c)=0. (Hint: Use the Extreme Value Theorem and Fermat's Theorem.)
If you must use Fermat's Theorem, then you need to show that there is a local extremum of f in (a,b). If you could show that, then you could invoke Fermat's Theorem, and you'd be done. How could you show that there is a local extremum in (a,b)?