Let a<b be elements of R, and let f,g:[a,b]-->R be continuous on [a,b] and differentiable on ]a,b[. Assume that f(a) is less than or equal to g(a) and that f'(x)<g'(x) for x element of ]a,b[. Show that f(x)<g(x) for x element of ]a,b].

Can this be done using Generalized Mean Value Theorem? And either way, could somebody give me some pointers?

Thanks a bunch!