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 $c \neq 0$, then $f$ "( $(c) \neq 0$
(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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