Question

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

Answer 1

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.