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Let f:R rarr R be a function satisfying ...

Let `f:R rarr R` be a function satisfying `f(x+y)=f(x)+lambda xy+3x^(2)y^(2)` for all `x, y in R`. If `f(3)=4 and f(5)=52` then f'(x) is equal to

A

10x

B

`-10x`

C

20x

D

128x

Text Solution

Verified by Experts

The correct Answer is:
B
We have
`f(x+y)=f(x)+lambdaxy+3x^(2)y^(2)"for all "x,y in R.......(i)`
Putting x=3 and y=2, we get
`f(5)=f(3)+6lambda+108`
`Rightarrow 52=4+6lambda+108`
`Rightarrow f(x+y)=f(x)-10xy+3xy^(2)y^(2)`
`Rightarrow (f(x+y)-f(x))/(y)=-10x+3x^(2)y`
`Rightarrow underset(y to 0)lim (f(x+y)-f(x))/(y)=underset(y to 0)lim -10x+3x^(2)y`
/`Rightarrow f'(x)=-10x`
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