If `f(x)` is a twice differentiable function and given that `f(1)=2,f(2)=5` and `f(3)=10` then

Options

A. `f'' (x)=2AAx in(1,3)`
B. `f''(x)=f'(x)=2` for some `x in (2,3)`
C. `f'' (x)=3 a x in (2,3)`
D. `f''(x)=2` for some `x in (1,3)`

Answer

D

Explanation

Let `g(x)=f(x)-x^(2)-1`
`g(1)=g(2)=0`
So, `g' (C_(1))=0` for at least one `C_(1)in (1,2)`
Similarly `g' (C_(2))=0` for at least one `C_(2) in (2,3)`
Using LMVT on `x in (C_(1),C_(2))`
`g'' (C)=(g'(C_(2))-g' (C_(1)))/(C_(2)-C_(1))=0`
`f'' (C_(3))=2`