Question

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

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

Answer 1

Answer: (A)

$f^{\prime }\left(x\right)=\frac{1}{1+x}-\frac{\left(2+x\right)2-2x-1}{{\left(2+x\right)}^2}$
$=\frac{1}{1+x}-\frac{4}{{\left(2+x\right)}^2}$
$=\frac{4+x^2+4x-4-4x}{\left(1+x\right){\left(2+x\right)}^2}$
$=\frac{x^2}{\left(1+x\right){\left(2+x\right)}^2}>0$
[if $1+x>0$ i.e., if $x-1$]