Question

How do you find the max or minimum of `f(x)=x^2-x-6`?

Answer 1

The minimum of `f(x)` is at `(1/2,-25/4)`

Explanation:

Let 's find the derivative of `f(x)`

`f(x)=x^2-x-6`

`f'(x)=2x-1`

`f'(x)=0` when `x=1/2`

Let's do a chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `1/2` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `darr` `color(white)(aaaa)` `uarr`

So it's a minimum.

We can also calculate the second derivative

`f''(x)=2>0`, so it'a a minimum

graph{x^2-x-6 [-14.24, 14.23, -7.12, 7.12]}