Question

Show that the function $f\left(x\right)=\frac{x+1}{x+2},x>0$ is increasing.

Answer 1

Given,

$f\left(x\right)=\frac{x+1}{x+2},x>0$.

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

or, $f^{\prime }\left(x\right)=\frac{1}{(x+2{)}^2}>0$ for all $x>0$.

So the function $f\left(x\right)$ is increasing for $x>0$.