Question

Find the intervals of monotonicity of the following functions.
$f\left(x\right)=x+\frac{1}{x+1}$

Answer 1

$f\prime \left(x\right)=1-(x+1{)}^{-2}f\prime (x)\ge 01-\frac{1}{{(1+x)}^2}\ge 01\ge \frac{1}{{(1+x)}^2}1+x^2+2x\ge 1x^2+2x\ge 0x(x+2)\ge 0x\in (-\infty ,-2\left)U\right(0,\infty )1-\frac{1}{(1+{x)}^2}\le 01+x^2+2x-1\le 0x(x+2)\le 0x\in (0,2)$