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Math Help - Word problem with differentiation..

  1. #1
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    Word problem with differentiation..

    The manager of a supermarket usually adds a markup of 20% to the wholesale prices of all the goods he sells. He reckons that he has a loyal core of F customers and that, if he lowers his markup to x% he will attract an extra k(20-x) customers from his rivals. Each week the average shopper buys goods whose wholesale value is $A.

    Show that the manager can increase his profit by reducing his mark-up below 20% provided that 20k > F.

    Now I am not sure of how to show for the bolded part in the question.

    Workings:

    .... So, as the mark up is lowered to x%:
    Let total customers C = F + 20k - kx
    Then total profit:

    $ x/100 * A(F + 20k - kx)
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  2. #2
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    If profit P=.01(AF+20Ak)x - .01(Ak)x^2,
    then the derivative P'=.01(AF+20Ak) - .02(Ak)x.
    Set P'=0 and x=(AF+20Ak)/2(Ak) = F/2k + 10, which is the optimal markup.
    If 20k > F, then F/2k < 10, so Xoptimal=F/2k + 10 < 20.
    QED
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