2007 Putnam B2

Suppose that $f: [0,1] \to \mathbb{R}$ has a continuous derivative

and that $\int_0^1 f(x)\,dx = 0$. Prove that for every $\alpha \in (0,1)$,

$$

\left| \int_0^\alpha f(x)\,dx \right| \leq \frac{1}{8} \max_{0 \leq x

\leq 1} |f'(x)|.

$$

假设 $f: [0,1] \to \mathbb{R}$ 具有连续导数

$\int_0^1 f(x)\,dx = 0$。证明对于每个 $\alpha \in (0,1)$，

$$

\左| \int_0^\alpha f(x)\,dx \right| \leq \frac{1}{8} \max_{0 \leq x

\leq 1} |f'(x)|。

$$