Question

For all twice differentiable functions $f:R\to R$, with $f\left(0\right)=f\left(1\right)=f^{\prime }\left(0\right)=0$,

• A. $f^{\prime \prime }\left(x\right)\ne 0$, at every point $x\in (0,1)$
• B. $f^{\prime \prime }\left(0\right)=0$
• C. $f^{\prime \prime }\left(x\right)=0$, for some $x\in (0,1)$
• D. $f^{\prime \prime }\left(x\right)=0$, at every point $x\in (0,1)$

Answer 1

The correct option is C $f^{\prime \prime }\left(x\right)=0$, for some $x\in (0,1)$

Applying rolle`s theorem in $[0,1]$ for function $f\left(x\right)$
$f^{\prime }\left(c\right)=0,c\in (0,1)$
again applying rolles theorem in $[0,c]$ for function $f^{\prime }\left(x\right)$
$\Rightarrow$ $f^{\prime \prime }\left(c_1\right)=0,c_1\in (0,c)$