How do you find the intervals of increasing and decreasing using the first derivative given y=x-2cosx?

Nov 10, 2017

The intervals of increasing are $\left(- \frac{1}{6} \pi + 2 k \pi , \frac{7}{6} \pi + 2 k \pi\right)$
The intervals of decreasing are $\left(\frac{7}{6} \pi + 2 k \pi , \frac{11}{6} \pi + 2 k \pi\right)$, $\forall k \in \mathbb{Z}$

Explanation:

Calculate the first derivative

$y = x - 2 \cos x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 1 + 2 \sin x$

The critical points are when

$\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$1 + 2 \sin x = 0$

$\sin x = - \frac{1}{2}$

$x \in \left(- \frac{1}{6} \pi + 2 k \pi\right) \cup \left(\frac{7}{6} \pi + 2 k \pi\right)$, $\forall k \in \mathbb{Z}$

We build a sign chart in the interval $x \in \left[- \frac{1}{6} \pi , \frac{19}{6} \pi\right]$

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \frac{1}{6} \pi$$\textcolor{w h i t e}{a a a a a a a}$$\frac{7}{6} \pi$$\textcolor{w h i t e}{a a a a a}$$\frac{11}{6} \pi$$\textcolor{w h i t e}{a a a a}$$\frac{19}{6} \pi$

$\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}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a a}$$-$$\textcolor{w h i t e}{a a}$$0$$\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}$color(white)(aaaa)↗$\textcolor{w h i t e}{a a}$color(white)(aaa)↘$\textcolor{w h i t e}{a a}$color(white)(aaaa)↗

Therefore,

The intervals of increasing are $\left(- \frac{1}{6} \pi + 2 k \pi , \frac{7}{6} \pi + 2 k \pi\right)$

The intervals of decreasing are $\left(\frac{7}{6} \pi + 2 k \pi , \frac{11}{6} \pi + 2 k \pi\right)$

$\forall k \in \mathbb{Z}$

graph{x-2cosx [-14.95, 17.09, -4.82, 11.2]}