Question

How do you find the exact relative maximum and minimum of the polynomial function of `f(x) = –2x^3 + 6x^2 + 18x –18`?

Answer 1

The relative maximum is `=(3,36)` and the relative minimum is `=(-1,-28)`

Explanation:

The polynomial function is

`f(x)=-2x^3+6x^2+18x-18`

This function is defined, continuous and derivable on `RR`

The first derivative is

`f'(x)=-6x^2+12x+18`

The critical points are when `f'(x)=0`

`-6(x^2-2x-3)=0`

`(x+1)(x-3)=0`

`f'(x)=-(x+1)(x-3)`

Therefore,

`x=-1` and `x=3`

We can make a variation chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-1` `color(white)(aaaa)` `3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+1` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `↘` `color(white)(aaaa)` `↗` `color(white)(aaa)` `↘`

The second derivative is

`f''(x)=-12x+12`

`f''(-1)=12+12=24`, `=>`, this is a relative minimum as `f''(-1)>0`

`f''(3)=-12*3+12=-24`, `=>`, this is a relative maximum as `f''(-1)<0`

graph{-2x^3+6x^2+18x-18 [-74.1, 74.1, -37.03, 37]}