The exercise is about convex functions:
How to prove that $f(t)=\int_0^t g(s)ds$ is convex in $(a,b)$ whenever $0\in (a,b)$ and $g$ is increasing in $[a,b]$?
I proved that
$$f(x)\leq \frac{x-a'}{b'-x}f(b')+\left(1-\frac{x-a'}{b'-x}\right)f(a')$$
when we have
$$x=\left(1-\frac{x-a'}{b'-a'}\right)a'+\frac{x-a'}{b'-a'}b'$$