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Let f:RtoR be a function such that f(x)=...

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

A

30

B

8

C

2

D

`-4`

Text Solution

Verified by Experts

The correct Answer is:
C
`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
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