Question

How do you use the first and second derivatives to sketch `y = x^3 - 12x - 12`?

Answer 1

See the answer bellow.

Explanation:

`y=x^3-12x-12`
First derivative: `y'=3x^2-12`
Second derivative: `y''=6x`

Using `y'=3x^2-12`
`3(x^2-4)`
`3(x-2)(x+2)`
`x=-2,2`
Sub x values into `y=x^3-12x-12`, to find y values
`y=(-2)^3-12(-2)-12`
`y=4`, `(-2,4)`
`y=(2)^3-12(2)-12`
`y=-28`, `(2,-28)`

Using `y''=6x`
Sub x values into `y''=6x`
`6(-2)=-12`
`x<0`
`:.` `(2,-28)` is a maximum point
`6(2)=12`
`x>0`
`:.` `(-2,4)` is a minimum point

When `x=0`
`y=0^3-12(0)-12`
`y=-12`
`(0,-12)`

Use the information from the first and second derivative to sketch the graph as well as the y intercept.

graph{y=x^3-12x-12 [-10, 10, -5, 5]}