Let $f:[a,b]\rightarrow R$ a monotone function (lets suppose increasing for the discussion)
Prove that exist point $c \in [a,b]$ such that:
$\int_{a}^{b} f \,dx = f(a)(c-a)+f(b)(b-c)$
I was able to prove that $f(a)(b-a)≤\int_{a}^{b} f\,dx≤f(b)(b-a)$
with Riemann sums both got stuck, note that $f$ is not necessary continues so the IVT theorem can't be used.