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

Feb 6, 2017

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 $f \left(x\right) = - {x}^{2} - 8 x - 12$

$f ' \left(x\right) = - 2 x - 8$

Critical point when $f ' \left(x\right) = 0$

$- 2 x - 8 = 0$, $\implies$, $x = - 4$

Let construct the sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 4$$\textcolor{w h i t e}{a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$- 2 x - 8$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f ' \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a}$$0$$\textcolor{w h i t e}{a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a a}$↗$\textcolor{w h i t e}{a}$$0$$\textcolor{w h i t e}{a a}$↘

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

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