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(C) Given that $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with $f(a)=0$ and $f(b)=0$. By Rolle's theorem, there exists at least one

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$f^{\prime \prime}(x) = 0$ for some $x \in(a, b)$

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(C) Given that $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ with $f(a)=0$ and $f(b)=0$. By Rolle's theorem, there exists at least one $c \in (a, b)$ such that $f^{\prime}(c)=0$. Since $f^{\prime}(a)=0$ and $f^{\prime}(c)=0$, and $f^{\prime}$ is continuous on $[a, c]$ and differentiable on $(a, c)$, we can apply Rolle's theorem to $f^{\prime}$ on the interval $[a, c]$. Therefore, there exists at least one $k \in (a, c)$ such that $f^{\prime \prime}(k)=0$. Since $(a, c) \subset (a, b)$, there exists $x \in (a, b)$ such that $f^{\prime \prime}(x)=0$.

Let $f$ be any continuously differentiable function on $[a, b]$ and twice differentiable on $(a, b)$ such that $f(a)=f^{\prime}(a)=0$ and $f(b)=0$. Then:

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