Question

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a twice continuously differentiable function such that \(f(0)=f(1)=f^{\prime}(0)=0\). Then
(A) \(f^{\prime \prime}(0)=0\)
(B) \(f^{\prime \prime}(c)=0\) for some \(c \in \mathbb{R}\)
(C) if \(\mathrm{c} \neq 0\), then \(\mathrm{f}^{\prime \prime}(\mathrm{c}) \neq 0\)
(D) \(f^{\prime}(x)>0\) for all \(x \neq 0\)

Answer 1

Ans: (B)
Hint : \(f(0)=f(1)=0\)
\(\Rightarrow f^{\prime}(k)=0\) for same \(k \in(0,1)\) by Rolle's Theorem.
Again, \(f^{\prime}(0)=f^{\prime}(k)=0\)
\(\Rightarrow \mathrm{f}^{\prime \prime}(\mathrm{c})=0\) for same \(\mathrm{c} \in(0, \mathrm{k})\) by Rolle's Theorem.

Answer 2

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