Let $f\in C^2([a,b])$ such that $f''\ge k \ \forall k \in \mathbb{R}$. Show that $\forall x \in [a,b]$: $\frac{f(x)-f(a)}{x-a}\le \frac{f(b)-f(a)}{b-a}-\frac{k}{2}(b-x)$.
My attempt was to use Taylor therem to approximate $f(x)$ around $a$ and then try to bound. But, I didn't really succeed and get a wanted result. If someone could give a small track how to prove the statement, I would really appreciate it. Thank you in advance for help!