Question 19
Let $f : [0, 1] \to \mathbb{R}$ be a differentiable function with $f(0) = f(1)$, $\int_{0}^{1} f(x) dx = 0$ and $f'(x) \neq 1$, for every $x \in [0, 1]$.
a) Prove that the function $g : [0, 1] \to \mathbb{R}$ given by $g(x) = f(x) - x$ is decreasing.
b) Prove that for any integer $n \ge 1$ the following inequality holds
$$
\left| \sum_{k=0}^{n-1} f \left( \frac{k}{n} \right) \right| \lt \frac{1}{2}.
$$
a) Prove that the function $g : [0, 1] \to \mathbb{R}$ given by $g(x) = f(x) - x$ is decreasing.
b) Prove that for any integer $n \ge 1$ the following inequality holds
$$
\left| \sum_{k=0}^{n-1} f \left( \frac{k}{n} \right) \right| \lt \frac{1}{2}.
$$