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.)
Are you locked into using the EVT and Fermat's Theorem? A more intuitive approach would be to try to use the Intermediate Value Theorem on f'(x). The only catch there is that I'm not sure you know that f' is continuous.
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)?