The manager of a large apartment complex knows from experience that 120 units will be occupied if the rent is 450 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 5 dollar increase in rent. Similarly, one additional unit will be occupied for each 5 dollar decrease in rent. What rent should the manager charge to maximize revenue?
q(p) = 120 - (p-450) = 570 - p
revenue= p * q(p) = p (570 - p)
maximumrevenue when d(revenue)/dp = 0
0= (570 - p) * p = -2p + 570
p= 285 is a critical point.
d^2(revenue)/dp^2= -2 < 0 : so p=285 is a maximum.