calculus question?

Question

can a piece-wise function posess extrema on an interval?

Answer 1

Yes, it can.
Example,
y = 0, x < -1
y = |x|, -1 ≦ x < 1
y = 1, x ≧ 1

Answer 2

Yes. You could have a fuction such as
y = -x [- infinity,0)
y = x^2 [0 , infinity]
This would have a minimum at x =0.
You could add on as many pieces as you want and have several relative max and min points.

Answer 3

yes