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
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]}