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

1 Answer
Sep 19, 2016

Answer:

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.