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