MathDB

Let

f: [0,1]\rightarrow \mathbb{R}

a derivable function such that f(0)\equal{}f(1), \int_0^1f(x)dx\equal{}0 and

f^{\prime}(x) \neq 1\ ,\ (\forall)x\in [0,1]

. i)Prove that the function g: [0,1]\rightarrow \mathbb{R}\ ,\ g(x)\equal{}f(x)\minus{}x is strictly decreasing. ii)Prove that for each integer number

n\ge 1

, we have: \left|\sum_{k\equal{}0}^{n\minus{}1}f\left(\frac{k}{n}\right)\right|<\frac{1}{2}

Problem Statement