Question

How do you find all critical point and determine the min, max and inflection given `f(x)=x^4-4x^3+20`?

Answer 1

Please see the explanation below

Explanation:

Calculate the first and second derivatives

The function is

`f(x)=x^4-4x^3+20`

Calculate the first derivative

`f'(x)=4x^3-12x^2`

`f'(x)=0`

`=>`, `4x^3-12x^2=0`

`=>`, `4x^2(x-3)=0`

The critical points are `x=0` and `x=3`

Construct a variation chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `0` `color(white)(aaaaaa)` `3` `color(white)(aaaa)` `+oo`

`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)(aaaa)` `↗`

There is a local minimum at `(3, -7)`

Calculate the second derivative

`f''(x)=12x^2-24x`

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

`12x^2-24x=0`

`=>`, `12x(x-2)=0`

`=>`, `x=0` and `x=2`

The inflection points are `(0, 20)` and `(2,4)`

Build a variation chart to determine the concavities

`color(white)(aaaa)` `" Interval "` `color(white)(aaaa)` `(-oo, 0)` `color(white)(aaaa)` `(0,2)` `color(white)(aaaa)` `(2,+oo)`

`color(white)(aaaa)` `" sign f''(x)"` `color(white)(aaaaaaa)` `+` `color(white)(aaaaaaa)` `-` `color(white)(aaaaaaaa)` `+`

`color(white)(aaaa)` `" f(x)"` `color(white)(aaaaaaaaaaaa)` `uu` `color(white)(aaaaaaa)` `nn` `color(white)(aaaaaaaa)` `uu`

graph{x^4-4x^3+20 [-32.73, 32.24, -5.85, 26.6]}