Question

How do you find the intervals of increasing and decreasing given `y=-x^4+3x^2-3`?

Answer 1

The intervals of increasing are `x in (-oo,-1.225) uu (0,1.225)`.
The intervals of decreasing are `x in (-1.225,0) uu (1.225, +oo)`

Explanation:

We calculate the first derivative and build a chart.

`y=-x^4+3x^2-3`

`y'=-4x^3+6x`

We have, `y'=0` when

`-4x^3+6x=0`

`x(-4x^2+6)=0`

`x(6-4x^2)=0`

`x=0`,

and `x=+-sqrt(3/2)=+-1.225`

The chart is :

`color(white)(a)` `I` `color(white)(a)` `(-oo,-1.225)` `color(white)(a)` `(-1.225,0)` `color(white)(a)` `(0,1.225)` `color(white)(a)` `(1.225, +oo)`

`color(white)(a)` `y'` `color(white)(aaaaaa)` `+` `color(white)(aaaaaaaaaaa)` `-` `color(white)(aaaaaaa)` `+` `color(white)(aaaaaaaa)` `-`

`color(white)(a)` `y` `color(white)(aaaaaa)` `↗` `color(white)(aaaaaaaaaaa)` `↘` `color(white)(aaaaaaa)` `↗` `color(white)(aaaaaaaa)` `↘`