Mr. David manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost, in rupees, of producing x units is 240 + bx + cx2, where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66.66%. However, an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit.
How many units should Mr. David produce daily ?	Choose the most appropriate option:	A) 130	B) 100	C) 70	D) 150	E) None of these
Explanation :   Cost price function f(x) = cx2 + bx + 240.   When x = 20, f(20) = 400c + 20b + 240.   =&gt; f(40) = 1600c + 40b + 240   66.67% is 2/3.   =&gt; f(40) – f(20) = 2/3 * f(20).   Substituting f(40) and f(20), we get, 1200c+ 20b =(2/3) * (400c + 20b + 240)   Solving, we get 20b + 2800c = 480.   Now, f(60) = 3600c + 60b + 240.   f(60) – f(40) = ½ * f(40)   Substituting, and solving, we get, 360 + 60b + 2400c = 240 + 60b + 3600c   c = 1/10.   Substituting we get, b = 10.   So, the cost function f(x) = 0.1x2 + 10x + 240.   Selling price = 30 * x. ( Each unit SP is 30)   So, profit = SP – CP = 30x – (0.1x2 + 10x + 240) = -x2/10 + 20x – 240.   For maximum profit, differentiating this should give 0.   dp/dt = 0, dp/dt = -x/5 + 20   20 – x/5 = 0. x= 100.   Also double differentiating d2p / dt2, we get a negative number, so profit is maximum.   So, profit is maximum when 100 units are produced daily.   Answer : 100
<result>B</result>
Mr. David manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost, in rupees, of producing x units is 240 + bx + cx2, where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66.66%. However, an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit.
How many units should Mr. David produce daily ?