Note: this is not a typical cubic function

The average revenue for product Q is given by AR = 200 – 3.5Q and the total cost of Q


STC = 500-6Q+3Q^2 < Note: this is not a typical cubic function

a.    Find the price function and then the TR function. Hint: another name for price is average revenue (AR).

b.   Write the MR and MC functions below. Remember: MR = dTR/dQ and MC = dSTC/dQ. 

c.    What positive value of Q will maximize total profit? Remember, letting MR = MC signals the objective of total profit maximization. Solve MR = MC for Q.  The value of Q you get should not be zero or negative)?

d.   Use the price function found in (a) to determine the price per unit that will need to be charged at the Q found in (c). This will be the price you should ask for the total profit maximizing quantity.

e.    What total profit will result from selling the quantity found in (c) at the price found in (d)? Hint: the profit function is found as TR – STC.

f.     At what level of Q is revenue maximized?  Remember, let MR = 0 and solve for Q. MR = 0 signals the objective of maximizing revenue.

g.   At what level of Q is average profit per unit maximized?  Hint: the average profit function is the total profit function found in (e) divided by Q. To find the level of Q that maximizes average profit, find the first derivative of average profit, set this derivative equal to zero and solve for Q.

h.   What price per unit should be charged at the quantity found in (g)? Simply plug the Q you got in (g) into the same price function you found in (a) and also used in (d).

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