Question

If $f(x)$ is a twice differentiable function and given that $f(1)=1,f(2)=4,f(3)=9$, then

• A. $f^{\prime \prime }(x)=2$, for $\forall x\in (1,3)$
• B. $f^{\prime \prime }(x)=f^{\prime }(x)=2$, for some $x\in (2,3)$
• C. $f^{\prime \prime }(x)=3$, for $\forall x\in (2,3)$
• D. $f^{\prime \prime }(x)=2$, for some $x\in (1,3)$

Answer 1

Answer: (D)

$f(1)=1,f(2)=4,f(3)=9$
$\text{ Let }g(x)=f(x)-x^2$
$\text{ We have }g(1)=0,g(2)=0,g(3)=0$
hence by Rolles Theorem $g^{\prime }(x)=0$ for some $c\in (1,2)$ are $g^{\prime }(x)=0$ for some $d\in (2,3)$ again using Rolles Theorem

for $g^{\prime }(x)=f^{\prime }(x)-2x$
$f^{\prime \prime }(x)=0\text{ for some }x\in (c,d)$
$f^{\prime \prime }(x)-2=0\text{ for some }x\in (c,d)$
$f^{\prime \prime }(x)=2\text{ for some }x\in (1,3)$