# Is #f(x)=(x^3+3x^2-x-9)/(x+1)# increasing or decreasing at #x=-2#?

##### 1 Answer

Feb 25, 2016

Increasing.

#### Explanation:

To determine if a function is increasing or decreasing at a point, examine the sign of the first derivative at that point.

- If
#f'(-2)<0# , then#f(x)# is decreasing at#x=-2# . - If
#f'(-2)>0# , then#f(x)# is increasing at#x=-2# .

To find **quotient rule.**

This gives us:

#f'(x)=((x+1)d/dx(x^3+3x^2-x-9)-(x^3+3x^2-x-9)d/dx(x+1))/(x+1)^2#

Find each derivative through the **power rule.**

#f'(x)=((x+1)(3x^2+6x-1)-(x^3+3x^2-x-9))/(x+1)^2#

Simplification of the numerator yields:

#f'(x)=(2x^3+6x^2+6x+8)/(x+1)^2#

Find the sign of the derivative at

#f'(-2)=(2(-8)+6(4)+6(-2)+8)/(-2+1)^2=4#

Since

graph{(x^3+3x^2-x-9)/(x+1) [-7.87, 6.18, -1.176, 5.85]}