Maharashtra State BoardHSC Commerce 12th Board Exam
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The total cost of manufacturing x articles is C = 47x + 300x2 - x4. Find x, for which average cost is increasing - Mathematics and Statistics

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Sum

The total cost of manufacturing x articles is C = 47x + 300x2 - x4.  Find x, for which average cost is increasing

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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 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. 

Concept: Increasing and Decreasing Functions
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