Let `f:[1,oo]` be a differentiable function such that `f(1)=2.` If `int_1^xf(t)dt=3xf(x)-x^3` for all `xgeq1,` then the value of `f(2)` is

Options

A. 3
B. 4
C. 5
D. 6

Answer

D

Explanation

We have,
`6int_(1)^(x)f(t)dt=3xf(x)-x^(3)`
Differentiating both sides with respect to x, we get
`6f(x)=3f(x)+3xf'(x)-3x^(2)`
`rArr" "3f(x)=3xf'(x)-3x^(2)`
`rArr" "(d)/(dx)(f(x))+(-(1)/(x))f(x)=x`
This is a linear differential equation with integrating factor
`e^(int-(1)/(x)dx)=e^(-logx)=(1)/(x)`
Solution is given by
`(1)/(x)f(x)=intx xx(1)/(x)dx+C`
`rArr" "f(x)=x^(2)+Cx`
`therefore" "f(1)=1+CrArr2=1+CrArrC=1`
Thus, `f(x)=x^(2)+x`
`therefore" "f(2)=4+2=6`