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

Answer

B

Explanation

We have,
`f(x)=x^(n)`
`implies" "f'(x)=nx^(n-1)`
`:." "f'(alpha+beta)=f'(alpha)+f'(beta)" ""for all "alpha,betagt0`
`implies" "n(alpha+beta)^(n-1)=nalpha^(n-1)+nbeta^(n-1)" ""for all "alpha,betagt0`
`implies" "(alpha+beta)^(n-1)=alpha^(n-1)+beta^(n-1)" ""for all "alpha,betagt0`
`implies" "n-1=1impliesn=2`
Also, for `n=0` we have
`f(x)=1" ""for all "x`
`implies" "f'(x)=0" ""for all "x`
`:." "f'(alpha+beta)=f'(alpha)+f'(beta)`
Hence, `n=0,2.`