Question

$f\left(x\right)=x-\log \left(\frac{x+1}{x}\right)(x>0)$ is increasing in

• A. $(0,\infty )$
• B. (1, 2)
• C. $(1,2)U(2,\infty )$
• D. $(-\infty ,\infty )$

Answer 1

The correct option is A $(0,\infty )$

$f\left(x\right)=x-\log \frac{x+1}{x}$
$f^{\prime }\left(x\right)=\frac{x^2+x+1}{x(x+1)}$
For f(x) to be increasing,
$f^{\prime }\left(x\right)>0$
$\Rightarrow \frac{x^2+x+1}{x(x+1)}>0$
$\Rightarrow x>0$
$\Rightarrow x\in (0,\infty )$