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f:RrarrR,f(x)=x^(3)+x^(2)f'(1)+xf''(2)+f...

`f:RrarrR,f(x)=x^(3)+x^(2)f'(1)+xf''(2)+f'''(3)" for all "x in R.`
The value of f(1) is

A

2

B

3

C

-1

D

4

Text Solution

Verified by Experts

`"Here, "f(x)=x^(3)+x^(2)f'(1)+xf''(2)+f'''(3)`
`"Put "f'(1)=a,f''(2)=b,f'''(3)=c" (1)"`
`therefore" "f(x)=x^(3)+ax^(2)+bx+c`
`"or "f'(x)=3x^(2)+2ax+b`
`"or "f'(1)=3+2a+b" (2)"`
`"or "f''(x)=6x+2a`
`"Also, "f''(2)=12+2a" (3)"`
`"or "f'''(x)=6`
`"Further "f'''(3)=6" (4)"`
`"From "(1) and (4), c=6.`
`"From "(1),(2) and (3),` we have
`a=-5,b=2`
`thereforef(x)=x^(3)-5x^(2)+2x+6`
`"or "f'(x)=3x^(2)-10x+2`
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