Question

What are the critical values, if any, of `f(x) = -3x^2+x-1`?

Answer 1

The critical value is a maximum at `(1/6,-11/12)`

Explanation:

We must differentiate and do a sign chart

We use

`(x^n)'=nx^(n-1)`

`f(x)=-3x^2+x-1`

`f'(x)=-6x+1`

`f'(x)=0`, when `-6x+1=0`, `=>`, `x=1/6`

`f(1/6)=-3/36+1/6-1=-11/12`

We can do our sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaaa)` `1/6` `color(white)(aaaaa)` `+oo`

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

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `uarr` `color(white)(aa)` `-11/12` `color(white)(aaa)` `darr`

The critical value is a maximum at `(1/6,-11/12)`