Question 83
Let $f : [0, 1] \to \mathbb{R}$ be a continuous function with $\int_{0}^{1} f(x)dx = \int_{0}^{1} x f(x)dx$ and $f(0) = 0$. Prove that there exists $c \in (0, 1)$ so that
$$
\int_{0}^{c} x f(x)dx = \frac{c}{2} \int_{0}^{c} f(x)dx.
$$
$$
\int_{0}^{c} x f(x)dx = \frac{c}{2} \int_{0}^{c} f(x)dx.
$$