Question

How do you use the first derivative to determine where the function `f(x)= 3 x^4 + 96 x` is increasing or decreasing?

Answer 1

`f(x)` is decreasing when `x in ] -oo,-2 ]`
`f(x)` is increasing when `x in[-2, +oo[`

Explanation:

`f(x)=3x^4+96x`

The derivative is,
`f'(x)=12x^3+96`

`f'(x)=0`

when, `12x^3+96=0`

`12x^3=-96`

`x^3=-8`

`x=-2`

Let's do a sign chart,

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaaaa)` `-2` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `f'(x)` `color(white)(aaaaaa)` `-` `color(white)(aaaaa)` `0` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `darr` `color(white)(aaa)` `-144` `color(white)(aaaa)` `uarr`

So, `f(x)` is decreasing when `x in ] -oo,-2 ]`

and `f(x)` is increasing when `x in[-2, +oo[`