Maximizing Prots. Now we look at the prot of a rm which equals to the revenue minus the cost. i.e. if 13(5)) denotes the revenue when :1: units are produced and sold and C(m) represents the cost, then the prot of selling 2: units equals to 11(1) 2 RM?) C(33) It is assumed that the second derivatives of both R(m) and C(33) exist and are continuous. Usually, the goal of a rm is to maximize the prot. We will discuss some interesting consequences when prot is maximized by using the First and Second Derivative Tests. Question 3. (a) Prove that if the prot attains a local maximum at :00 > 0, then R'(:r:0) : (7(a)) and R\"(mo) S C\"($o). (b) In a competitive market, the unit price of a commodity is a constant 190. Write down, in this case, the revenue function R(:r:). Deduce that when the prot is maximized, the marginal cost equals to p0. (c) Suppose that for a certain commodity, . when a units are demanded, the unit price is p(x) = -0.0032x + 10, . the cost of producing x units is C(x) = 4000 + 2x - 0.0012x2. Write down its revenue function R(x) and its profit function II(x). Then find the maximum profit