Question

How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for `f(x)=(x^3)/(x^2-4)`?

Answer 1

The intervals of increasing are `x in (-oo,-sqrt12) uu(sqrt12, +oo)`
The intervals of decreasing are `x in (-sqrt12,-2)uu(-2,+2)uu(2,sqrt12)`

Explanation:

We calculate the first derivative and construct a sign chart.

We need

`(u/v)'=(u'v-uv')/(v^2)`

`f(x)=x^3/(x^2-4)`

`u=x^3`, `=>`, `u'=3x^2`

`v=x^2-4`, `=>`, `v'=2x`

`f'(x)=(3x^2(x^2-4)-2x*x^3)/(x^2-4)^2`

`=(3x^4-12x^2-2x^4)/(x^2-4)^2`

`=(x^4-12x^2)/(x^2-4)^2`

`=(x^2(x^2-12))/(x^2-4)^2`

`f'(x)=(x^2(x+sqrt12)(x-sqrt12))/((x+2)^2(x-2)^2)`

We construct the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaaa)` `-oo` `color(white)(aaaa)` `-sqrt12` `color(white)(aaa)` `-2` `color(white)(aaa)` `0` `color(white)(aaa)` `2` `color(white)(aaa)` `sqrt12` `color(white)(aaa)` `+oo`

`color(white)(aaaa)` `x+sqrt12` `color(white)(aaaaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaa)` `+` `color(white)(a)` `+` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+2` `color(white)(aaaaaaaaaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaa)` `+` `color(white)(a)` `+` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-2` `color(white)(aaaaaaaaaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaa)` `+` `color(white)(a)` `+` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-sqrt12` `color(white)(aaaaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaa)` `-` `color(white)(a)` `-` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f'(x)` `color(white)(aaaaaaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaa)` `-` `color(white)(a)` `-` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaaaa)` `↗` `color(white)(aaaa)` `↘` `color(white)(aaa)` `↘` `color(white)(a)` `↘` `color(white)(aa)` `↘` `color(white)(aaaa)` `↗`

The relative maximum is when `x=-sqrt12`

The relative minimum is when `x= sqrt12`

graph{x^3/(x^2-4) [-14.24, 14.24, -7.12, 7.12]}