Question

Find the intervals in which the function $f\left(x\right)=\log (1+x)-\frac{2x}{2+x}$ is increasing or decreasing.

Answer 1

from the given function if we write domain it becomes;

for $\log x$ condition is $x>0$ from this we get

$(1+x)>0$

$x>-1$

from $\frac{2x}{2+x}$

we get $(2+x)=0$ that is $x=-2$ is not in our domain

so our domain is $x>-1$

for checking increasing or decreasing we need $f^{\prime }\left(x\right)$

$f^{\prime }\left(x\right)=\frac{1}{1+x}-\frac{2(2+x)-2x\left(1\right)}{(2+x{)}^2}$

$f^{\prime }\left(x\right)=(\frac{1}{1+x})\times (\frac{x}{2+x}{)}^2$

we know that any squared expression is greater than or equal to zero;

$(\frac{x}{2+x}{)}^2>0$,equality occurs when $x=0$

and from domain $(1+x)>0$ which also means $\frac{1}{1+x}>0$

so from this we can say that $f^{\prime }\left(x\right)>0$ for all $x>-1$,except for $x=0$ it becomes $0$.

so therefore we conclude that $f\left(x\right)$ increases in the intervals $xϵ(-1,0)$ and $xϵ(0,\infty )$