How to prove $r(x)>p(x)$? Given two functions $r(x)$ and $p(x)$, both of which are defined on closed interval $x\in[a,b]$. 
Functions $r(x)$ and $p(x)$ also satisfy the following constraints:
\begin{cases}
r(a)=p(a)\\
r'(x)>p'(x)
\end{cases}
Then how to prove $r(x)>p(x)$ on a left open and right closed interval $(a,b]$?

Let $h(x)=r(x)-p(x)$, then $h'(x)>0$ so $h$ is strictly increasing. As $h(a)=0$, we have $h(x)>0$ for $x>a$.
