Let $f:[0,1]\rightarrow \mathbb{R}$ be a continuously differentiable function such that $$\int_0^1 f(x) \, dx=0$$ and $m \leq f'(x) \leq M$ on $(0,1)$. Prove that $$\frac{m}{12} \leq \int_0^1 xf(x) \, dx \leq \frac{M}{12}.$$
This is just the last bonus question in our test yesterday. I wasn't able to answer of course. Though I did verify it by letting $\displaystyle f(x)= \left( x−\frac{1}{2} \right)^3$ for which I found $0\le f′(x)\le 3/4$ on $(0,1)$ and $$0 \le \int^1_0 xf(x) \, dx=\frac{1}{80} \le \frac{3}{48}.$$
I do not know how to prove this.