Assum that f and g are differentiable on the interval (-c, c) and f(0) = g(0).
show that if f'(x) > g'(x) for all x (0,c), then f(x) > g(x) for all x (0,c).
Assume there is such that then by Rolle's theorem there exists a point such that which is a contradiction since in and this also implies that in a right neighbourhood of we have that is positive. And this is enough because since is continous, if it were negative there would exist a point in which it's zero and we repeat the argument. So is positive in