Question

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

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

Answer 1

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

Applying Rolle's theorem to $f\left(x\right)$ on the interval $[0,1]$, we get
$f^{\prime }\left(c_1\right)=0$ for some $c_1\in (0,1)$

Again, applying Rolle's theorem to $f^{\prime }\left(x\right)$ on the interval $[0,c_1]$, we get
$f^{\prime \prime }\left(c\right)=0$ for some $c\in (0,c_1)$