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The cost accountant of a firm producing colour television has worked out the total cost function for the firm as TC=120Q-Q^2+0.02Q^3. A sales manager has provided the sales forecasting function as P=114-0.25Q .Where P is price and Q is the quantity sold. Required, 1. Find the level of production that will yield minimum average cost per unit and determine whether this level of output maximizes profit for the firm. 2. Determine the price that will maximize profit for the firm. 3.Determine the maximum revenue for this firm.

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  1. #1. avg cost is just TC/Q. So, the avg cost A is
    A = 120-Q+0.02Q^2
    minimum A is just the vertex of that parabola.

    revenue = price * quantity = PQ
    R = (114-0.25Q)Q = 114Q - 0.25Q^2

    profit = revenue - cost, so the profit
    y = (114Q - 0.25Q^2)-(120Q-Q^2+0.02Q^3) = -0.02Q^3 + 0.75Q^2 - 6Q

    maximum revenue/profit is, of course, where
    dR/dQ = 0 and dy/dQ = 0

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