Question

Suppose $f$ and $g$ are differentiable functions on $(0,\infty )$ such that $f\text{'}\left(x\right)=-\frac{g\left(x\right)}{x}$ and $g\text{'}\left(x\right)=-\frac{f\left(x\right)}{x}$, for all $x>0$. Further, $f\left(1\right)=3$ and $g\left(1\right)=-1$.

If $f\left(x\right)+g\left(x\right)=A{x}^k,$ for all $x>0$ and some constant $A$, then the value of $k$ equals

• A. $1$
• B. $2$
• C. $-1$
• D. $-2$

Answer 1

The correct option is C $-1$

Differentiating,
$\frac{d}{dx}\left\{x\right[\left(f\right(x)+g(x\left)\right]\}=f(x)+g(x)+x[f^{\prime }\left(x\right)+g\left(x\right)]=f(x)+g(x)-g(x)-f(x)=0\Rightarrow x[f\left(x\right)+g\left(x\right)]=A\Rightarrow f(x)+g(x)=A{x}^{-1}\Rightarrow k=-1$