I am trying to solve the following problem:
Let $f:[0,1]\to\mathbb{R}$ a continuous function such that $$\int_{0}^{1}f(x)dx=0.$$ Prove that there exists $c\in(0,1)$ such that $$\int_{0}^{c}xf(x)dx=0.$$
I guess that it is necessary some version of the Mean Value Theorem to prove this statement.
If $F(x)=\int_{0}^{x}f(t)dt$, we know by the Rolle's Theorem that $F'(\alpha)=f(\alpha)=0$ for some $\alpha\in(0,1)$. However, I have not found any interesting use for this $\alpha$.
If $G(x)=\int_{0}^{x}tf(t)dt$, we have to prove that $G(c)=0$ for some $c\in (0,1)$. If we assume that it is false, we have by the Intermediate Value Theorem that either $G(x)>0$ for all $x\in(0,1)$ or $G(x)<0$ for all $x\in(0,1)$.
Would you give me some ideas to complete the proof?
