Let $f(x) =x^n$, n being a non-negative integer. The value of n for which the equality f'(x + y) = f'(x) + f'(y) is valid for all x, y > 0 is
Answer & explanation
Correct answer: option 4
We have,
$f(x)=x^n \Rightarrow f(x+y)=(x+y)^n \Rightarrow f'(x+y)=n(x+y)^{n-1}$
Also, $f'(x)=n x^{n-1}$ and $f'(y)=n y^{n-1}$
∴ $f'(x+y)=f'(x)+f'(y)$
$\Rightarrow n(x+y)^{n-1}=n . x^{n-1}+n . y^{n-1}$
$\Rightarrow (x+y)^{n-1}=x^{n-1}+y^{n-1}$
Clearly, this is true for n = 2 only.