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Question
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Solution
Given,
Total cost function is (C) = 47x + 300x2 – x4
Average cost CA = `"C"/"A"`
∴ CA = `(47 x + 300x^2 – x^4)/x`
∴ CA = `(x(47 + 300x – x^3))/x`
∴ CA = 47 + 300x – x3
`"dC"_"A"/"dx" = "d"/"dx" 47 + 300x – x^3`
∴ `"dC"_"A"/"dx"` = 0 + 300 – 3x2
∴ `"dC"_"A"/"dx"` = 3(100 – x2)
Since average cost, CA is an increasing function, `"dC"_"A"/"dx" > 0`
∴ 3(100 – x2) > 0
∴ 100 – x2 > 0
∴ 100 > x2
∴ x2 < 100
∴ – 10 < x < 10
∴ x > – 10 and x < 10
But x > – 10 is not possible. ...[∵ x > 0]
∴ x < 10
∴ The average cost CA is increasing for x < 10.
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