Question

The function $f\left(x\right)=\log (1+x)-\frac{2x}{2+x}$ is increasing on

• A. $(0,\infty )$
• B. $(-\infty ,0)$
• C. $(-\infty ,\infty )$
• D. None of the above

Answer 1

Answer: (A)

$(0,\infty )$

Given $f\left(x\right)=\log (1+x)-\frac{2x}{2+x}$
$\therefore$ $f^{\prime }\left(x\right)=\frac{1}{1+x}-\frac{(2+x).2-2x}{{(2+x)}^2}$
$=\frac{x^2}{(1+x){(x+2)}^2}$
Clearly $f^{\prime }\left(x\right)>0$ for all $x>0$
Hence $f\left(x\right)$ is increasing on $(0,\infty )$