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Question
The marginal cost of production of a firm is given by C'(x) = `20 + x/20` the marginal revenue is given by R’(x) = 30 and the fixed cost is ₹ 100. Find the profit function
Solution
C'(x) = `20 + x/20`
Fixed cost k1 = ₹ 100
C(x) = `int "C"_1 (x) "d"x + "k"_1`
C = `int (20 + x/20) "d"x +"k"_1`
= `20x + x^2/40 + "k"_1`
= `20x + x^2/40 + 100`
R'(x) = 30
R(x) = `int "R'"(x) "d"x + "k"_2`
R = `int 30 "d"x + "k"_2`
R = `30 "d"x+ "k"_2`
When x = 0
R = 0
⇒ k2 = 0
∴ R = 30x
Profit function P = `"R" - "C"`
= `(30x) - (20x + x^2/40 + 100)`
∴ P = `10x - x^2/10 - 100`
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