How do you find the critical numbers for f(x) = |x + 3| - 1 to determine the maximum and minimum?

1 Answer
Jim H
Sep 19, 2016

The only critical number is -3. (And f(-3)=1 is a minimum.)

Explanation:

The derivative of f(x) = absx is

f'(x) ={ (-1," if",x<0),(1," if",x>0) :}
(and does not exist at x=0).

f(x) = abs(x+3)-1 is absx translated 3 left and down 1.

f'(x) is { (-1," if",x " is left of the vertex"),(1," if",x " is right of the vertex") :}
and is undefined at the vertex.
The vertex is at x = -3.

I must admit, this seems like an awful lot of work to find the minimum and maximum.
We know that absx has a "V" shaped graph, so a translation also has a "V" shaped graph with minimum at the vertex and no maximum.

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