Question 83
The function $f : [0, 1] \to \mathbb{R}$ is differentiable and
$$
\int_{0}^{1} f(x) \, dx = \int_{0}^{1} x f(x) \, dx = 0.
$$
Prove that there exists $c \in (0, 1)$ such that $f'(c) = 0$.
$$
\int_{0}^{1} f(x) \, dx = \int_{0}^{1} x f(x) \, dx = 0.
$$
Prove that there exists $c \in (0, 1)$ such that $f'(c) = 0$.