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Question
A manufacturing company produces x items at the total cost of Rs (180 + 4x). The demand function of this product is P = (240 − x). Find x for which profit is increasing.
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Solution
Total cost function (C) = 180 + 4x
Demand function (P) = 240 − x
Where x is the number of items produced.
Total revenue (R) = P × D
∴ R = x (240 − x)
∴ R = 240x − x2
Profit function π = R − C
∴ π = (240x − x2) − (180 + 4x)
∴ π = 240x − x2 − 180 − 4x
∴ π = − x2 + 236x − 180
Differentiating w.r.t.x,
∴ `"dπ"/"dx"` = − 2x + 236
Profit π is increasing if `"dπ"/"dx"` > 0
i.e. if − 2x + 236 > 0
i.e. if 236 > 2x
i.e. if x < `236/2`
i.e. if x < 118
∴ The profit is increasing for x < 118.
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The manufacturing company produces x items at the total cost of ₹ 180 + 4x. The demand function for this product is P = (240 − 𝑥). Find x for which profit is increasing
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∴ `("dR")/("d"x) = square`
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