Question

How do you find the max or minimum of `f(x)=3x^2-7x+2`?

Answer 1

There is a minimum at `(7/6,-25/12)`

Explanation:

We can factorise `f(x)=(3x-1)(x-2)`

We calculate the derivative of `f(x)` and any critical point is found when `f'(x)=0`

`f(x)=3x^2-7x+2`

`f'(x)=6x-7`

`f'(x)=0`, when `6x-7=0`. `=>`, `x=7/6`

We make a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaa)` `1/3` `color(white)(aaaaa)` `7/6` `color(white)(aaaaaa)` `2` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` #color(white)(aaa)↘# `color(white)(aa)` `-25/12` `color(white)(aaaaaa)` `↗` `color(white)(aa)`

There is a minimum at `(7/6,-25/12)`

We can also calculate `f''(x)=6`

`f''(x)>0`, we have a minimum