Assume $f(x)$ has continuous derivative on $[0,1]$ ,$|f'(x)| \leq M$ Assume $f(x)$ has continuous derivative on $[0,1]$ with $|f'(x)| \leq M$ and
$f(0)=f(1)=0$.

Prove that 
$$\left| \int_0^1 f(x) \ dx \right| \leq \frac{M}{4}$$
A hint: Try to imagine how fast $f$ can grow from the ends towards the center of the interval.

By the way, you can replace $\leq$ by $&lt;$ in the claim.
Hint: Let $g(x)=x-a$ where $a=\frac{3 + \sqrt{6}}{6}$. Using the Cauchy Inequality
$$ \left(\int_0^1f'(x)g(x)dx\right)^2\le\int_0^1(f'(x))^2dx\int_0^1(g(x))^2dx $$
I think you will get the desired inequality. Note that
$$ \int_0^1(x-a)^2dx=\frac{1}{4}, \int_0^1(x-a)f'(x)dx=-\int_0^1f(x)dx. $$