Suppose f '' ( x ) exists for all real x . If f ( 2 ) = 2 ,   f ( 3 ) = 5 and f ( 4 ) = 10 , then which…

Consider

$f\left(x\right)=a{x}^2+bx+c$

$f\left(2\right)=4a+2b+c=2 \dots \left(\mathrm{I}\right)$

$f\left(3\right)=9a+3b+c=5 \dots \left(\mathrm{II}\right)$

$f\left(4\right)=16a+4b+c=10 \dots \left(\mathrm{III}\right)$

Subtract equation $\left(\mathrm{I}\right)$ from equation $\left(\mathrm{II}\right)$,

$5a+b=3 \dots \left(\mathrm{IV}\right)$

Subtract equation $\left(\mathrm{II}\right)$ from equation $\left(\mathrm{III}\right)$,

$7a+b=5 \dots \left(\mathrm{V}\right)$

Subtract equation $\left(\mathrm{IV}\right)$ from equation $\left(\mathrm{V}\right)$,

$2a=2$

$a=1$

Substitute the value in equation $\left(\mathrm{IV}\right)$,

$5\times 1+b=3$

$b=-2$

Now,

$f^{\text{'}}\left(x\right)=2x-2$

$f^{\text{'}\text{'}}\left(x\right)=2$

Since, $f^{\text{'}\text{'}}\left(x\right)>1$ for some $x\in (2,4)$.