Question

Let $f$ be twice differentiable function satisfying $f\left(1\right)=1,f\left(2\right)=4,f\left(3\right)=9$, then

• A. $f^{\prime \prime }\left(x\right)=2,\forall xϵR$
• B. $f^{\prime }\left(x\right)=5=f^{\prime \prime }\left(x\right)$. For some $xϵ(1,3)$
• C. There exists atleast one $xϵ(1,3)$ such that $f^{\prime \prime }\left(x\right)=2$
• D. None of the above

Answer 1

The correct option is C There exists atleast one $xϵ(1,3)$ such that $f^{\prime \prime }\left(x\right)=2$

Let $g\left(x\right)=f\left(x\right)-x^2$
$\Rightarrow g\left(x\right)$ has atleast $3$ real roots which are $x=1,2,3$ (by mean value theorem)
$\Rightarrow g^{\prime }\left(x\right)$ has atleast $2$ real roots in $xϵ(1,3)$
$\Rightarrow g^{\prime \prime }\left(x\right)$ has atleast $1$ real root in $xϵ(1,3)$
$\Rightarrow f^{\prime \prime }\left(x\right)=2$ for atleast one $xϵ(1,3)$.