Question

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

Answer 1

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

`(1/x)'=-1/x^2`

The domain of `y` is `D_y=RR-{0}`

We calculate the first derivative

`y=x+4/x`

`dy/dx=1-4/x^2`

To find the critical points, we calculate the values of `x` when `dy/dx=0`

when

`1-4/x^2=0`

`1=4/x^2`

`x^2=4`

Therefore, `x=-2` and `x=2`

We can build the chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaaaaaa)` `0` `color(white)(aaaaaaa)` `2` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `dy/dx` `color(white)(aaaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `y` `color(white)(aaaaaaaaaa)` `↗` `color(white)(aaaa)` `↘` `color(white)(aaa)` `||` `color(white)(aaa)` `↘` `color(white)(aaaa)` `↗`

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

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