Question

Find the maxima and minima, if any, for the function f given by f(x)=x^4-14x^2+24x+6. Also find the values of the function at those points.?

Answer 1

The local max. is at `=(1,17)` and the local min. are at `=(-3, -111)` and `=(2,14)`

Explanation:

Calculate the first derivative and draw a variation chart

`f(x)=x^4-14x^2+24x+6`

`f'(x)=4x^3-28x+24=4(x^3-7x+6)`

Determine the roots of `f'(x)`

`f'(1)=4-28+24=0`

Therefore, `(x-1)` is a factor of `4x^3-28x+24`

Perform a long division,

`(x^3-7x+6)/(x-1)=x^2+x-6=(x-2)(x+3)`

Therefore,

`f'(x)=4(x-1)(x-2)(x+3)`

The critical points are `x=1`, `x=2` and `x=-3`

The variation chart is as follows :

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-3` `color(white)(aaaaaaa)` `1` `color(white)(aaaaaaa)` `2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `0` `color(white)(aaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x-1` `color(white)(aaaaa)` `-` `color(white)(aaaa)` ###color(white)(aaaa)#`-` `color(white)(aa)` `0` `color(white)(aa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x-2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` ###color(white)(aaaa)#`-` `color(white)(a)` ###color(white)(aaaa)#`-` `color(white)(aa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f'(x)` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `0` `color(white)(aaa)` `+` `color(white)(aaa)` `0` `color(white)(a)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `↘` `color(white)(aa)` `-111` `color(white)(aa)` `↗` `color(white)(aa)` `17` `color(white)(a)` `↘` `color(white)(a)` `14` `color(white)(aa)` `↗`

graph{x^4-14x^2+24x+6 [0.837, 2.734, 19.631, 20.579]}