Let f: R rightarrow R be a twice continuously | vedclass

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(A) Given that $f(0) = f(1) = 0$. By Rolle's theorem, there exists at least one point $c \in (0, 1)$ such that $f^{\prime}(c) = 0$. We are also given

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

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

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

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