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Question
The cost function of a firm is C = x3 – 12x2 + 48x. Find the level of output (x > 0) at which average cost is minimum.
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Solution
The cost function is C = x3 – 12x2 + 48x
Average cost is minimum,
When Average Cost (AC) = Marginal Cost (MC)
Cost function, C = x3 – 12x2 + 48x
Average Cost, AC = `(x^3 - 12x^2 + 48x)/x` = x2 – 12x + 48
Marginal Cost (MC) = `"dC"/"dx"`
`= "d"/"dx" (x^3 - 12x^2 + 48x)`
= 3x2 – 24x + 48
But AC = MC
x2 – 12x + 48 = 3x2 – 24x + 48
x2 – 3x2 – 12x + 24x = 0
-2x2 + 12x = 0
Divide by -2 we get, x2 – 6x = 0
x (x – 6) = 0
x = 0 (or) x – 6 = 0
x = 0 (or) x = 6
But x > 0
∴ x = 6
Output = 6 units
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