SEEMOUS 2011 Problem 4

Let $f:[0,1]\to\mathbb R$ be a twice continuously differentiable increasing function. Define the sequences given by $L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right)$ and $U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right)$ , $n\ge1$ . 1. The interval $[L_n,U_n]$ is divided into three equal segments. Prove that, for large enough $n$ , the number $I=\int^1_0f(x)\text dx$ belongs to the middle one of these three segments.