At least I believe you need the Mean Value Theorem:

Let f:[0,1]->R and g:[0,1]->R be differentiable with f(0)=g(0) and f'(x)>g'(x) for x in [0,1]. Prove that for x in (0,1], we have that f(x)>g(x).

My professor set h(x)=f(x)-g(x) and then said that h(x)>0, h'(x)>0 and that is literally all he said to explain this problem. This doesn't really make much sense to me. He is never formal about anything. Is that even remotely correct?