Analyzing Functions from Derviatives.
a)What are the Critical points of f.
b)On what intervals is f increasing or decreasing.
Answer the following questions above about the function
f'(x)=(x^2(x-1))/(x+2), x is not equal to -2
Given `f'(x)=(x^2(x-1))/(x+2)` :
The critical points occur when the first derivative is zero or fails to exist. The first derivative fails to exist at x=-2, but that is not in the domain.
`==>x=0,x=1,x=-2` are the critical points.
We test values on the intervals `(-oo,-2),(-2,0),(0,1),(1,oo)` :
`f'(-3)=36>0` so the function is increasing on `(-oo,-2)`
`f'(-1)=-2<0` so the function is decreasing on `(-2,0)`
`f'(1/2)=-1/2<0` so thefunction is decreasing on (0,1)
`f'(2)=1>0` so the function is increasing on `(1,oo)`
The function has a local minimum at x=1 since it is decreasing from the left, and increasing to the right. There is no local maximum.