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Question
The total cost of producing x units is ₹ (x2 + 60x + 50) and the price is ₹ (180 − x) per unit. For what units is the profit maximum?
Solution
Given, no. of units = x,
selling price of each unit = ₹ (180 – x)
∴ selling price of x unit = ₹ (180 - x).x
= ₹ (180x - x2)
Also, cost price of x units = ₹ (x2 + 60x + 50)
Now, Profit = P = Selling price – Cost price
= 180x - x2 - (x2 + 60x + 50)
= 180x - x2 - x2 - 60x - 50
∴ P = - 2x2 + 120x - 50
∴ `"dP"/"dx" = - 4"x" + 120`
and `("d"^2"P")/"dx"^2 = - 4`
Consider, `"dP"/"dx" = 0`
∴ - 4x + 120 = 0
∴ - 4x = - 120
∴ x = 30
For x = 30,
`("d"^2"P")/"dx"^2` = - 4 < 0
∴ P, i.e. profit is maximum at x = 30.
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