# How do you find the intervals of increasing and decreasing using the first derivative given y=x+4/x?

Mar 17, 2017

The intervals of increasing are x in ]-oo,-2[uu]2,+oo[
The intervals of decreasing are x in ]-2,0[uu]0,2[

#### Explanation:

We need

$\left(\frac{1}{x}\right) ' = - \frac{1}{x} ^ 2$

The domain of $y$ is ${D}_{y} = \mathbb{R} - \left\{0\right\}$

We calculate the first derivative

$y = x + \frac{4}{x}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 1 - \frac{4}{x} ^ 2$

To find the critical points, we calculate the values of $x$ when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

when

$1 - \frac{4}{x} ^ 2 = 0$

$1 = \frac{4}{x} ^ 2$

${x}^{2} = 4$

Therefore, $x = - 2$ and $x = 2$

We can build the chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 2$$\textcolor{w h i t e}{a a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a}$$2$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 2$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 2$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$\frac{\mathrm{dy}}{\mathrm{dx}}$$\textcolor{w h i t e}{a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a a a}$↗$\textcolor{w h i t e}{a a a a}$↘$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a}$↘$\textcolor{w h i t e}{a a a a}$↗

The intervals of increasing are x in ]-oo,-2[uu]2,+oo[

The intervals of decreasing are x in ]-2,0[uu]0,2[