Let `f : RtoR` be a function such that `f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3),x in R`. Then f(2) equals

Options

A. 30
B. 8
C. 2
D. `-4`

Answer

C

Explanation

`f(x)=x^(3)+x^(2)f'(1)+xf"(2)+f"'(3)`
f"(3)=6
`f'(x)=3x^(2)+2xf'(1)+f"(2)`
`f'(1)=3+2f'(1)+f"(2)`
`f'(1)+f"(2)+3=0` …(i)
`f"(x)=6x+2f'(1)`
f"(2)=12+2f'(1) …(ii)
Substitute in (i)
`f'(1)+12+2f'(1)+3=0`
`rArrf'(1)=-5f"(2)=2`
`f(x)=x^(3)-5x^(2)+2x+6`
f(2)=2