Question

Question #5faa3

Answer 1

I'm going to assume you mean the open intervals
for which
`f(x)` is increasing, and
`f(x)` is decreasing.
(The open interval of `f(x)` is (`- oo, + oo`)

If `f(x) =12x - x^3`
then the critical points (where the function changes from increasing to decreasing, or visa versa) occur when
`(d f(x))/(dx) = 0`

`(d (12x - x^3))/(dx) = 12 - 3x^2`

So the critical points occur when
`3x^2 = 12` or `x = +- 2`

When `x = -2`
`f(x) = 12(2) - (-2)^3 = 32`

When `x = 2`
`f(x) = 12(2) - (2)^3 = 16`

Therefore, for `x` in the open interval (`-2,+2`), f(x) is decreasing.

It follows that for `x` in the open intervals (`-oo,-2`) and (`2,+oo`), f(x) is increasing.

Hope this is what you were looking for.