Assume that all of the functions are twice differentiable and the second derivatives are never 0. 66.

Assume that all of the functions are twice differentiable and the second derivatives are never 0.

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66. (a) If f and g are concave upward on I, show that f + g is con- cave upward on I. (b) If f is positive and concave upward on I, show that the function g(x) = [f(x)]* is concave upward on I. 67. (a) If f and g are positive, increasing, concave upward func- tions on I, show that the product function fg is concave upward on I. (b) Show that part (a) remains true if f and g are both decreasing. (c) Suppose f is increasing and g 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? 68. Suppose f and g are both concave upward on (-, c). Under what condition onf will the composite function h(x) = f(g(x)) be concave upward?