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The total cost of manufacturing x articles is C = 47x + 300x^{2} − x^{4}. Find x, for which average cost is increasing.

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#### Solution

Given,

Total cost function is (C) = 47x + 300x^{2} – x^{4}

Average cost C_{A} = `"C"/"A"`

∴ C_{A} = `(47 x + 300x^2 – x^4)/x`

∴ C_{A} = `(x(47 + 300x – x^3))/x`

∴ C_{A} = 47 + 300x – x^{3}

`"dC"_"A"/"dx" = "d"/"dx" 47 + 300x – x^3`

∴ `"dC"_"A"/"dx"` = 0 + 300 – 3x^{2}

∴ `"dC"_"A"/"dx"` = 3(100 – x^{2})

Since average cost, C_{A} is an increasing function, `"dC"_"A"/"dx" > 0`

∴ 3(100 – x^{2}) > 0

∴ 100 – x^{2} > 0

∴ 100 > x^{2}

∴ x^{2} < 100

∴ – 10 < x < 10

∴ x > – 10 and x < 10

But x > – 10 is not possible. ...[∵ x > 0]

∴ x < 10

∴ The average cost C_{A} is increasing for x < 10.

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