Question

Find the interval of increase and decrease of the following functions.
$f\left(x\right)=2x+\frac{2}{x}$

Answer 1

$f\left(x\right)=2x+\frac{2}{x}$ where $x\ne 0$

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

For function to be increasing-

$f^{\prime }\left(x\right)>0$

$⟹2-\frac{2}{x^2}>0$

$⟹\frac{1}{x^2}<1$

$⟹x^2>1$

$⟹x\in (-\infty ,-1)\cup (1,\infty )$

For function to be decreasing-

$⟹2-\frac{2}{x^2}<0$

$⟹\frac{1}{x^2}>1$

$⟹x^2<1$

$⟹x\in (-1,1)$