Question 529847: 5. 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
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 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
---
P'(x) = -x^2+10x+11
---
P''(x) = -2x+10
----
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):
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Cheers,
Stan H.
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