RO-TST 2014 Problem 19

Let $n$ a positive integer and let $f\colon [0,1] \to \mathbb{R}$ an increasing function. Find the value of : $$\max_{0\leq x_1\leq\cdots\leq x_n\leq 1}\sum_{k=1}^{n}f\left ( \left | x_k-\frac{2k-1}{2n} \right | \right )$$