# Analyzing Functions from Derviatives. a)What are the Critical points of f. b)On what intervals is f increasing or decreasing. c)local max/minAnswer the following questions above about the function...

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.