Question

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

• A. $f^{\prime }\left(0\right)=0$
• B. $f^{\prime \prime }\left(c\right)=0$ for some $c\in R$
• C. if $c\ne 0$, then $f^{\prime \prime }\left(c\right)=0$
• D. $f^{\prime }\left(x\right)>0$ for all $x\ne 0$

Answer 1

The correct option is B $f^{\prime \prime }\left(c\right)=0$ for some $c\in R$

$f\left(x\right)$ is continuous and differentiable
$f\left(0\right)=f\left(1\right)=0\Rightarrow$ by rolle's theorem
$f^{\prime }\left(a\right)=0,$ $a$ belongs to (0,1)
given $f^{\prime }\left(0\right)=0$
by rolles theorem
$f^{\prime \prime }\left(0\right)=0$ for some $c$ , $c$ belongs to $(0,a)$
Therefore Answer is $B$