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

Options

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

Answer

C

Explanation

We have,
`f(x)= x^(3) + x^(2) f'(1) + xf''(2)+f'''(3)`
`rArr f'(x) = 3x^(2)+2xf'(1) + f''(2)" "`...(i)
`rArr f''(x) = 6x+2f'(1)" "...(ii)
`rArr f'''(x) = 6" "...(iii)
` rArr f'''(3) = 6 `
Putting x = 1 in Eq. (i), we get
` f'(1) = 3 + 2 f'(1) + f''(2)` " "...(iv)
and putting x = 2 in Eq. (ii), we get
` f''(2) = 12 + 2 f'(1)` ...(v)
From Eqs. (iv) and (v) , we get
` f'(1) = 3+2f'(1)+(12+2f'(1))`
` rArr" " 3f'(1) =- 15`
` rArr" " f'(1) =- 5`
` rArr" " f''(2) = 12 + 2 (-5) = 2" "` [using Eq. (v) ]
`:." " f(x) = x^(3) + x^(2) f'(1)+xf''(2) + f'''(3)`
` rArr" " f(x) = x^(3) - 5x^(2) + 2x+6`
` rArr" " f(2) = 2^(3) - 5(2)^(2) + 2(2) + 6=8- 20 + 4+6 =- 2`