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.