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.

Posted on

## We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

- 30,000+ book summaries
- 20% study tools discount
- Ad-free content
- PDF downloads
- 300,000+ answers
- 5-star customer support

Already a member? Log in here.

Are you a teacher? Sign up now