Question

How do you find the maximum or minimum of `f(x)=-7-3x^2+12x`?

Answer 1

The maximum is at `(2,5)`

Explanation:

We calculate the derivative
`f(x)=-7-3x^2+12x`
The domain of `f(x)` is `RR`
`f'(x)=-6x+12`
For a maximum or minimum, `f'(x)=0`
so, `-6x+12=0` `=>` `6x=12`
`x=2`
To see if it's a max or min, we make a chart
`x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `2` `color(white)(aaaa)` `+oo`
`f'(x)` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`
`f(x)` `color(white)(aaaaa)` `uarr` `color(white)(aaaa)` `darr`

So we have a maximum at `(2,5)`
we can also look at the sign of the second derivative
`f''(x)=-6` `(< 0)` this is a max
graph{-3x^2+12x-7 [-12.66, 12.65, -6.33, 6.33]}