How to integrate some function-inequality?

Question

I'm concerned with the subject of integrating function inequalities, namely given a function $r\in C^{1,\alpha}([0,s_{max}];\mathbb{R})$ and a constant $A$ satisfying the ineqality

$\begin{align} \frac{r'(s_2)-r'(s_1)}{s_2-s_1}\leq A \ \ \text{for all} \ \ s_1<s_2. \end{align}$ (1)

Now it's the task to gain the inequality

$\begin{align} r(s_2)-r(s_1)\leq r'(s_1)(s_2-s_1)+\frac{1}{2}A(s_2-s_1)^2 \end{align}$

by integration.

At first I'm not sure which should be the variable to integrate in (1), moreover (1) suggests there are only constants. The next is the way of integration I'm struggling with. I hope someone get it by the first view and can give me some hint :)

Thank you

Answer 1

Rewrite the equation as $r'(x) \le A(x - s_1) + r'(s_1)$ and integrate it from $s_1$ to $s_2$ with respect to $x$.