`f(x) = |x + 3| - 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 32 - Calculus of a Single Variable (10th Edition, Ron Larson).
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embizze | High School Teacher | (Level 2) Educator Emeritus

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Given the function f(x)=|x+3|-1:

Rewrite as a piecewise function:

`f(x)={ [[x+2,x > - 3],[-1,x=-3],[-x-4,x<-3]] `

The function is continuous on the reals.

The first derivative fails to exist at x=-3, and is nonzero for all other x's. (f'(x)=1 on `(-oo,-3) ` , and f'(x)=-1 on `(-3,oo) ` .)

Thus the only critical value is at x=-3. The function increases for x>-3 (since the first derivative is positive on this interval) and decreases for x<-3 (since the first derivative is negative on this interval.)

There is an absolute minimum at x=-3.

The graph:

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