The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f,g, and h whose values at 0 are all 0 and, for x ‰  0.

(a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0.
(b) Show that has neither a local maximum nor a local minimum at 0, has a local minimum, and has a local maximum.

/(x) = r'sin- g(x) 2sin