`f(x) = (x^2 - 2x + 1)/(x + 1)` Find the critical numbers, open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.

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Chapter 3, 3.3 - Problem 36 - Calculus of a Single Variable (10th Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Given: `f(x)=(x^2-2x+1)/(x+1)`

Find the critical numbers by setting the first derivative equal to zero and solving for the x value(s).

`f'(x)=[(x+1)(2x-2)-(x^2-2x+1)(1)]/(x+1)^2=0`

`2x^2-2x+2x-2-x^2+2x-1=0`

`x^2+2x-3=0`

`(x+3)(x-1)=0`

`x=-3,x=1`

The critical numbers are x=-3 and x=1. Critical numbers also exist where f(x) is not defined. Therefore x=-1 is also a critical number.

If f'(x)>0 the function is increasing on the interval.

If f'(x)<0 the function is decreasing on the interval.

Choose an x value that is less than -3.

f'(-4)=.5556 Since f'(-4)>0 the function is increasing in the interval

(-`oo,-3).`

Choose an x value that is between -3 and -1.

f'(-2)=-3 Since f'(-2) >0 the function is decreasing in the interval (-3, -1).

Choose an x value that is between -1 and 1.

f'(0)=-3 Since f'(0)<0 the function is decreasing in the interval (-1, 1).

Choose an x value that is greater than 1.

f(2)=.5556 since f'(2)>0 the function is increasing in the interval (1, `oo).`

Since the function changed direction from increasing to decreasing there is a relative maximum at x=-3. The relative maximum is at the point (-3, -8).

Since the function changed direction form decreasing to increasing there is a relative minimum at x=1. The relative minimum is at the point (1, 0).

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