# integral inequality involving $\sup|f'|$

Let $f:[0,1]\rightarrow \mathbb R$ be continuous function differentiable on $(0,1)$ with property that there exists $a \in (0,1]$ such that

$$\int_{0}^a f(x)dx=0$$

Prove that

$$\left|\int_{0}^1 f(x)dx \right|\le \dfrac {1-a} 2 \cdot \sup_{x\in (0,1)} |f'(x)|$$

Find the case of equality.

We have one solution using Mean value theorem.Is there another one?

• The factor $1/2$ bothers me, any idea where it comes from? – Daniel Aug 22 '15 at 11:43