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(B) Given that $f(0)=0$ and $f(1)=0$. By Rolle's Theorem, there exists at least one $c_1 \in (0, 1)$ such that $f^{\prime}(c_1)=0$. We are also given

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$f^{\prime \prime}(c)=0$ for some $c \in (0, 1)$

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(B) Given that $f(0)=0$ and $f(1)=0$. By Rolle's Theorem, there exists at least one $c_1 \in (0, 1)$ such that $f^{\prime}(c_1)=0$. We are also given $f^{\prime}(0)=0$. Now, consider the function $f^{\prime}(x)$ on the interval $[0, c_1]$. Since $f$ is twice continuously differentiable, $f^{\prime}$ is continuous on $[0, c_1]$ and differentiable on $(0, c_1)$. We have $f^{\prime}(0)=0$ and $f^{\prime}(c_1)=0$. By applying Rolle's Theorem to $f^{\prime}(x)$ on the interval $[0, c_1]$, there exists at least one $c \in (0, c_1)$ such that $f^{\prime \prime}(c)=0$. Since $(0, c_1) \subset (0, 1)$, there exists some $c \in (0, 1)$ such that $f^{\prime \prime}(c)=0$.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice continuously differentiable function such that $f(0)=f(1)=f^{\prime}(0)=0$. Then:

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