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Math Help - Demand and Total Cost Curves

  1. #1
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    Demand and Total Cost Curves

    I am given that:
    Q=100-P
    TC = 1/3Q-7Q+111Q+50

    Find output and price that maximise profit and then calculate profit. Ensure you have found a maximum and checked for boundary maxima.

    So far I have done this:

    Rearranged Q so that
    P= -Q+100

    Q*P = TR = -Q+100 (Q) = -Q+100Q

    Profit = TR-TC= -Q+100Q-1/3Q-7Q+111Q+50
    =1/3Q-6Q-11Q+50

    First Derivative = -Q-12Q-11
    (-Q-11)(Q+1)
    Therefore Q=11 or Q=-1

    Then Used Second Deriv to work out Min/Max:
    =-2Q-12
    When Q=11, Maximum
    When Q=-1, Maximum

    Now I am unsure If I have made a mistake up to this point however I sort of come stuck here. Does anyone know if I have made any mistakes and what the next step would be?

    My thought is that you cant have a negative quantity and therefore Q must = 11. When Q = 11, P=89

    and then i was just going to plug the figures into the Profit formula and see what i get?!
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  2. #2
    MHF Contributor
    Joined
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    Assuming your algebra is correct, You are right to discount Q=-11.

    a maximum can occur at either:

    Q=0
    Q=100 (corresponds to P=0)
    Or somewhere in between.

    Your differenciation has given you the maximum profit in the "somewhere in between" range. "Check for boundary maxima" means check that this gives higher profits than Q=0 or Q=100.
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