1) The zero function has a maximum and minimum at every point. If this is cheating, just take the zero function and give it two "spikes", one above the x axis and one below it.
3) Let . Chuse an R so large that for all , . Consequently, we must have . Since f is continuous, it achieves a maximum on the compact interval [-R,R]. The maximum must be at least , so since outside of [-R,R], it is less than , the maximum it achieves on [-R,R] is the maximum on all of .