CMA Foundation · Fundamentals of Business Mathematics and Statistics · Calculus - Application in Business
A Surat firm's average cost per unit is AC = x + 400/x + 20 (in ₹), where x is output. What output minimises average cost, and what is the minimum average cost?
Differentiate average cost: 1 − 400/x² equals zero when x = 20. The second derivative is positive, so it is a minimum. Substituting gives 20 + 20 + 20 = ₹60 per unit. Leaving out the constant term 20 would wrongly give ₹40.
- Ax = 20, minimum AC = ₹60Correct
- Bx = 20, minimum AC = ₹40
- Cx = 400, minimum AC = ₹421
- Dx = 10, minimum AC = ₹70
Explanation
d(AC)/dx = 1 − 400/x² = 0 gives x² = 400, x = 20. Second derivative 800/x³ > 0 so it is a minimum. AC = 20 + 20 + 20 = 60. The ₹40 option omits the constant 20.
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