Intersection and derivatives

Question

If two functions $f$ and $g$ are such that:

• $f(0) = g(0)$
• $\forall x > 0,\ f^\prime(x) > g^\prime(x)$

is it true that $f\neq g$ for any $x>0$?

I believe so, but I don't know how to prove it. This seems simple though, there might be something I don't see quite yet.

Answer 1

Let $h(x)=f(x)-g(x)$. Then $h(0)=0$ and $h'(x)>0$ for all $x>0$. If $h(a)=0$ for some $a>0$, we get a contradiction by applying the mean value theorem on the interval $[0,a]$.