Let $\displaystyle f: [0,1]\longrightarrow\Bbb{R}$ be a continous function and $\displaystyle f(0)=f(\frac{1}{2})=0$ . If $\displaystyle f$ be differentiable over $\displaystyle (0,1)$, prove that there exist $\displaystyle c\in (0,1)$ such that $\displaystyle \int_0^1\!f(x)\,dx\leqslant\frac{1}{4}|f'(c)|$.