Find the absolute maximum value of the function

The answer to this question is 1. If we differentiate each piece, we get
                       f'(x) =  {2,          -2 < x < 0
                                  { -2x,       x>0
The critical number is x = 0. We can also see that f(x) is decreasing on x>0.
We check the end points and critical numbers for the absolute maximum. One end point at
                                   f(-2) = 2(-2) + 1 = -3
The critical number checked gives
                                    f(0) = -0^2 + 1 = 1
The absolute maximum value is 1.