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25 March, 13:18

A soft-drink vendor at a popular beach analyzes his sales records and finds that if he sells x cans of soda pop in one day, his profit (in dollars) is given by P (x) = - 0.001x2 + 3x - 1850.

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  1. 25 March, 13:36
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    First derivative of the total profit function is

    dP/dx = d/dx (-0.001x^2 + 4x - 1850)

    = - 0.002x + 4

    When dP/dx is taken equal to 0

    - 0.002x + 4 =

    x = 4/0.002 = 2000

    Second derivative is

    d^2P/dx^2 = d/dx (dP/dx)

    = d/dx (-0.002x + 4)

    = - 0.002

    Since the second derivative is - ve, it can be said the profit function has a maximum at x = 2000

    Profit will be maximum when 2000 cans are sold

    1) Maximum profit = P (x=2000) = - 0.001 (2000) ^2 + 4*2000 - 1850

    = - 4000 + 8000 - 1850

    = 2150

    2) He must sell 2000 cans for getting maximum profit.
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