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Let f: R rarr R be a differentiable func...

Let `f: R rarr R` be a differentiable function satisfying `f(x+y)=f(x)+f(y)+x^(2)y+xy^(2)` for all real numbers x and y. If `lim_(xrarr0) (f(x))/(x)=1,` then
The value of f(9) is

A

240

B

356

C

252

D

730

Text Solution

Verified by Experts

The correct Answer is:
C
f(0)=0
`"So, "underset(xrarr0)lim(f(x))/(x)=f'(0)=t" (Usin L 'Hopital's rule)"`
`f(x+y)=f(x)+f(y)+x^(2)y+xy^(2)`
Differentiating w.r.t. x, keeping y constant, we get
`f'(x+y)=f'(x)+2xy+y^(2)`
`"Put "x=0. Then`
`f'(y)=y^(2)+1 or f'(x)=x^(2)+1`
`therefore" "f(x)=(x^(3))/(3)+x+c`
`f(0)=0=c`
`therefore" "f(x)=(x^(3))/(3)+x`
`f'(3)=10`
`"and "f(9)=243+9=252`
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