Assume that all of the functions are twice differentiable and the second derivatives are never 0.
(a) If f and are positive, increasing, concave upward functions on I, show that the product function fg is concave upward on I.
(b) Show that part (a) remains true if f and are both decreasing.
(c) Suppose f is increasing and is decreasing. Show, by giving three examples, that fg may be concave upward, concave downward, or linear. Why doesn't the argument in parts (a) and (b) work in this case?