Let `f(x)=x^n` ,`n` being a non negative integer. The value of `n` for which the equality`f'(a+b)=f'(a)+f'(b)` is valid for all `a.bgt0` is

Options

A. 0,1
B. 1,2
C. 2,4
D. none of these

Answer

D

Explanation

We have,
`f(x)=x^(n)impliesf(x+y)=(x+y)^(n)impliesf'(x+y)=n(x+y)^(n-1)`
Also, `f'(x)=nx^(n-1)" and "f'(y)=ny^(n-1)`
`:." "f'(x+y)=f'(x)+f'(y)`
`implies" "n(x+y)^(n-1)=n.x^(n-1)+n.y^(n-1)`
`implies" "(x+y)^(n-1)+x^(n-1)+y^(n-1)`
Clearly, this is true for n=2 only.