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Question
The total cost C for producing x units is Rs (x2 + 60x + 50) and the price is Rs (180 - x) per unit. For how many units the profit is maximum.
Solution
Revenue R = x (180 - x)
R = 180x - x2
Profit π(x) = R - C
= (180x - x2) - (x2 + 60x + 50)
π(x) = -2x2 + 120x - 50
`(dπ)/(dx)` = -4x + 120
Profit π is maximum
If `(dπ)/(dx)` = 0 and `(d^2y)/(dx^2)` < 0
`(dπ)/(dx)` = 0
-4x + 120 = 0
x = 30
Now, `(d^2π)/(dx^2)` = -4
`(d^2π)/(dx^2)` = -4 < 0at x = 30
Profit is maximum at x = 30 units.
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