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Question
The profit function P(x) of a firm, selling x items per day is given by P(x) = (150 – x)x – 1625 . Find the number of items the firm should manufacture to get maximum profit. Find the maximum profit.
Solution
Profit function P(x) is given by
P(x) = (150 – x)x – 1625
= 150x – x2 – 1625
∴ P'(x) = `"d"/("d"x)(150 x - x^2 - 1625)`
= 150 × 1 – 2x – 0
= 150 – 2x
and
P"(x) = `"d"/("d"x)(150 - 2x)`
= 0 – 2 × 1
= – 2
Now, P'(x) = 0 gives, 150 – 2x = 0
∴ x = 75
and
P''(75) = – 2 < 0
∴ by the second derivative test, P(x) is maximum when x = 75
Maximum profit = P(75)
= (150 – 75)75 – 1625
= 75 × 75 – 1625
= 4000
Hence, the profit will be maximum, if the manufacturer manufactures 75 items and maximum profit is 4000.
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