Question

A rm's total cost function is given as T C = 4 + 8Q − Q 2 , and total revenue as T R = 50Q − 4Q 2 , where Q is the number of units produced and sold (in thousands). How many units should be produced to maximize prot? 

Answer 1

Solution.

$$TC=4+8Q-Q^2 \newline TR=50Q-4Q^2$$

Profit = TR - TC = 50Q - 4Q² - (4 + 8Q - Q²) = 50Q - 4Q² - 4 - 8Q + Q² = -4 + 42Q - 3Q²

Profit has maximum than $\frac{d Profit}{d Q} = 0,\newline 42 - 3*2*Q=42-6Q=0.$From here Q = 7 (thousands)

Answer: 7