Question

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

Answer 1

The intervals of increasing are `(-1/6pi+2kpi, 7/6pi+2kpi)`
The intervals of decreasing are `(7/6pi+2kpi, 11/6pi+2kpi)`, `AA k in ZZ`

Explanation:

Calculate the first derivative

`y=x-2cosx`

`dy/dx=1+2sinx`

The critical points are when

`dy/dx=0`

`1+2sinx=0`

`sinx=-1/2`

`x in (-1/6pi+2kpi) uu (7/6pi+2kpi)`, `AA k in ZZ`

We build a sign chart in the interval `x in [-1/6pi, 19/6pi]`

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-1/6pi` `color(white)(aaaaaaa)` `7/6pi` `color(white)(aaaaa)` `11/6pi` `color(white)(aaaa)` `19/6pi`

`color(white)(aaaa)` `dy/dx` `color(white)(aaaaa)` `0` `color(white)(aaaa)` `+` `color(white)(aa)` `0` `color(white)(aaa)` `-` `color(white)(aa)` `0` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `y` `color(white)(aaaaaaa)` ###color(white)(aaaa)#`↗` `color(white)(aa)` ###color(white)(aaa)#`↘` `color(white)(aa)` ###color(white)(aaaa)#`↗`

Therefore,

The intervals of increasing are `(-1/6pi+2kpi, 7/6pi+2kpi)`

The intervals of decreasing are `(7/6pi+2kpi, 11/6pi+2kpi)`

`AA k in ZZ`

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