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Question
The average cost function, AC for a commodity is given by AC = `x + 5 + 36/x` in terms of output x. Find
1) The total cost, C and marginal cost, MC as a function of x.
2) The outputs for which AC increases
Solution
Let C be the total cost function. Then
Average cost (AC) = `C/x => AC.x => C = (x + 5 + 36/x) x`
`=> C = x^2 + 5x + 36`
Let MC be the marginal cost function. Then,
`MC = (dC)/dx = d/dx (x^2 + 5x + 36) = 2x + 5`
Now, `d/dx (AC) = d/dx (x + 5 + 36/x) = 1 - 36/x^2`
For AC to be increasing `d/dx(AC) > 0 => 1 - 36/x^2 > 0`
`=> x^2 - 36 > 0 => (x - 6) (x + 6) > 0`
`=> x < -6 or x > 6 => x > 6` [∵ x > 0]
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