(1) (15 %) Let $f : [a, b] \to \mathbb{R}$ be continuous and differentiable on $(a, b)$. Assume that $f(a) = 0$ and $\int_{a}^{b} f(x) \, dx = 0$. Prove that there is a $c \in (a, b)$ such that $f'(c) = 0$.
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(1) (15 %) Let $f : [a, b] \to \mathbb{R}$ be continuous and differentiable on $(a, b)$. Assume that $f(a) = 0$ and $\int_{a}^{b} f(x) \, dx = 0$. Prove that there is a $c \in (a, b)$ such that $f'(c) = 0$.