Analyzing Functions from Derviatives. a)What are the Critical points of f. b)On what intervals is f increasing or decreasing. c)local max/min Answer the following questions above about the function f'(x)=(x^2(x-1))/(x+2), x is not equal to -2   THANKS!!

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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.

`f'(x)=0 ==>x^2(x-1)=0`

`==>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.

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