Question

Let $f\left(x\right)=x^n$, $n$ being a non - negative integer . The number of values of $n$ for which $f^{\prime }(p+q)=f^{\prime }\left(p\right)+f^{\prime }\left(q\right)$ is valid for all $p,q>0$ is :

• A. $0$
• B. $1$
• C. $2$
• D. none

Answer 1

The correct option is C $2$

$f(p+q)=(p+q{)}^n$
$\Rightarrow f^{\prime }(p+q)=n(p+q{)}^{n-1}$
$\Rightarrow f^{\prime }\left(p\right)=n(p{)}^{n-1}$
$\Rightarrow f^{\prime }\left(q\right)=n(q{)}^{n-1}$
Using the binomial expansion we can see that the equation will be satisfied exactly $2$ times.