Question 529847
Find the maximum profit and the number of units that must be produced and sold in
order to yield the maximum profit. Assume that revenue, R(x), and cost, C(x), are in
thousands of dollars and x in thousands of units for
R(x) = 100x-x2
C(x) = 1/3 x^3-6x^2+89x+100

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Profit = Revenue - Cost
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P(x) = 100x-x^2-[(1/3)x^3-6x^2+89x+100]
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P(x) = (-1/3)x^3+5x^2+11x-100
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P'(x) = -x^2+10x+11
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P''(x) = -2x+10
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Solve: -x^2+10x+11 = 0
x^2-10x-11 = 0
(x-11)(x+1) = 0
x = -1 or x = 11
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P''(-1) = -2(-1)+10 = 12 (minimum at (-1,f(-1)))
P''(11) = -2(11)+10 = -12 (maximum at (11,f(11))
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Graph of P(x):
{{{graph(400,400,-10,50,-50,200,(-1/3)x^3+5x^2+11x-100)}}}
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Cheers,
Stan H.
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