Question

How do you find the intervals of increasing and decreasing using the first derivative given `y=-(x^2+8x+12)`?

Answer 1

The interval of increasing is `x in ]-oo, -4]`
The interval of decreasing is `x in [-4, +oo[`

Explanation:

We calculate the first derivative and then build a sign chart

Let `f(x)=-x^2-8x-12`

`f'(x)=-2x-8`

Critical point when `f'(x)=0`

`-2x-8=0`, `=>`, `x=-4`

Let construct the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaaa)` `-oo` `color(white)(aaaa)` `-4` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `-2x-8` `color(white)(aaaaaa)` `+` `color(white)(aa)` `0` `color(white)(aa)` `-`

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

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaaaa)` `↗` `color(white)(a)` `0` `color(white)(aa)` `↘`

The interval of increasing is `x in ]-oo, -4]`

The interval of decreasing is `x in [-4, +oo[`