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