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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.

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Best answer

Profit function P (x) is given by

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 the maximum profit is 4000.

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