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 $(a,b]$?

Simple Integration with $a>x\geq b$
$$r(x)=r(a)+\int_a^x r'(x) dx>p(a)+\int_a^x p'(x)dx=p(x)$$
