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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?
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Solution
Given: Number 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 = −4x + 120`
and `(d^2P)/dx^2 = -4`
∴ For maxima and minima,
Consider, `(dP)/dx = 0`
∴ 4x + 120 = 0
∴ −4x = −120
∴ x = 30
For x = 30,
`("d"^2"P")/"dx"^2` = −4 < 0
∴ Total profit is maximum when the number of units = 30.
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