Given the following total revenue TR and total cost TC functions for different firms, maximize profit r for the firms as follows: (1) set up the profit function π = TR — TC, (2) find the critical value(s) where π is at a relative extremum and test the second-order condition, and (3) calculate the maximum profit.

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(a) TR=440Q-302 (1)=440Q-302 - TC= 14Q+225 (140+225) =-3Q+426Q - 225 (2) '-6Q+426 = 0 Q = 71 critical value "=-6 <0 concave, relative maximum (3) 225 = 14, 898 =-3(71)+426(71) - (b) TR = 800Q-702 TC 2Q3 Q2 + 800 + 150 (1)=8000-70 - (20 - Q + 800 + 150) = -203-602 + 7200 - 150 =- (2)=-602-120 + 720 = 0 =-6(Q2+20 120) = 0 =-6(Q-10)(Q+12)=0 Q = 10 Q=-12 critical values (10.3) Taking the second derivative directly from (10.3), as explained in Problem 10.20 (c), and ignoring all negative critical values, "-12Q 12 "(10)=-12(10) 12-132 < 0 concave, relative maximum (3) =-2(10)-6(10) + 720(10) - 150 = 4450 (c) TR = 3200Q - 90, TC Q3 -1.502 +500 +425 (1) = === (2) = 3200Q 9Q - (Q - 1.5Q +500 +425) - =-Q3-7.50+31500-425 '=-302-15Q+3150 = 0 =-3(Q2+50-1050) = 0 =-3(Q-30)(Q+35) = 0 Q = 30 Q=-35 critical values "=-60-15 "(30)=-6(30) -15=-195 <0 concave, relative maximum (3)=(30)- 7.5(30) + 3150(30) - 425 = 60,325 (d) TR=500Q 11Q2, TC 303-202 +680 +175 = (1)=500Q 11Q - (303-202 +680 +175) = =-303-902 +432Q - 175