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 $f\left(x\right)-g\left(x\right)=B{x}^m,$ for all $x>0$ and some constant $B$, then the value of $m$ equals

• A. $1$
• B. $2$
• C. $1$
• D. $-2$

Answer 1

The correct option is A $1$

$\frac{d}{dx}\left[\frac{f\left(x\right)-g\left(x\right)}{x}\right]=\frac{x\left[f^{\prime }\right(x)-g^{\prime }(x\left)\right]-\left[f\right(x)-g(x\left)\right]}{x^2}=0\Rightarrow f\left(x\right)-g\left(x\right)=B{x}^1\Rightarrow m=1$