Question

How do you find the exact relative maximum and minimum of the polynomial function of `f(x)=2x^3+3x^2-12x`?

Answer 1

Relative maximum: `f(-2)=20`
Relative minimum: `f(1) = -9`

Explanation:

Given
`color(white)("XXX")f(x)=2x^3+3x^2-12x`

Note
`color(white)("XXX")`Relative minimums/maximums happen points where `f'(x)=0`

`f'(x) = 6x^2+6x-12`
`color(white)("XXX")=6(x^2+x-2)`
`color(white)("XXX")=6(x+2)(x-1)`

`f'(x)=0` for `x=-2` and `x=+1`
so these are the locations of the relative minimum/maximum values.

`f(-2) = -16+12+24=20`
`f(1) = 2+3-12 = -9`

Therefore
`color(white)("XXX")`the relative maximum is `20` (at `x=-2`)
`color(white)("XXX")`the relative minimum is `(-9)` ( at `x=1`)