CMA Foundation · Fundamentals of Business Mathematics and Statistics · Calculus - Application in Business
The total cost of producing x units is C(x) = 3x² + 12x + 192 (in rupees). At what output is the average cost per unit minimum?
Average cost is minimum at x = 8 units. Average cost is 3x + 12 + 192/x; setting its derivative 3 − 192/x² to zero gives x² = 64. At x = 8 average cost and marginal cost are both ₹60, which confirms the result.
- Ax = 6
- Bx = 8Correct
- Cx = 12
- Dx = 64
Explanation
Average cost AC = 3x + 12 + 192/x. Then dAC/dx = 3 − 192/x² = 0 gives x² = 64, so x = 8. The second derivative 384/x³ is positive, confirming a minimum. Equivalently, AC equals MC (6x + 12) at x = 8: 3(8) + 12 + 24 = 60 and 48 + 12 = 60.
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