Question

Let $f : R \to R$ be a twice continuously differentiable function such that $f(0) = f(1) = f'(0) = 0$. Then

• A. $f''(0) = 0$
• B. $f''(c) = 0$ for some $c \in R $
• C. if $c \neq 0$, then $f''(c) \neq 0$
• D. $f' (x) > 0$ for all $x \neq 0$

Answer 1

Answer: (B)

We have,
$f: R \rightarrow R$ be a twice continuously differentiable function such that
$f(0)=f(1)=f^{\prime}(0)=0$
Now, for atleast one value of $c_{1} \in(0,1)$,
$f^{\prime}\left(c_{1}\right)=0 $ (by Rolle's theorem)
Again, $f^{\prime}(0)=0=f^{\prime}\left(c_{1}\right)$
$\Rightarrow f^{\prime}(c)=0$ for some $c \in\left(0, c_{1}\right)$
(by Rolle's theorem)