Question

How do you find the intervals of increasing and decreasing using the first derivative given `y=(x+2)^(2/3)`?

Answer 1

The interval of decreasing is `x in ]-oo,-2[`
The interval of increasing is `x in ]-2,+oo[`

Explanation:

We need

`(x^n)=nx^(n-1)`

We calculate the first derivative

`y=(x+2)^(2/3)`

`dy/dx=2/3(x+2)^(-1/3)=2/3*1/(x+2)^(1/3)`

We can build the sign chart

`color(white)(aaaa)` `Interval` `color(white)(aaaa)` `]-oo,-2[` `color(white)(aaaa)` `]-2,+oo[`

`color(white)(aaaa)` `dy/dx` `color(white)(aaaaaaaaaaaa)` `-` `color(white)(aaaaaaaaaaaa)` `+`

`color(white)(aaaa)` `y` `color(white)(aaaaaaaaaaaaaa)` `↘` `color(white)(aaaaaaaaaaa)` `↗`