CMA Final · Strategic Cost Management · Business Application of Maxima and Minima
A firm's total cost function is C(x) = 2x² + 40x + 1,800 rupees, where x is the number of units produced. At what output is the average cost per unit minimum?
Average cost is minimised at 30 units. Differentiating AC = 2x + 40 + 1800/x and equating to zero gives x squared equal to 900. The second derivative is positive, confirming a minimum, and marginal cost equals average cost at that output.
- A20 units
- B30 unitsCorrect
- C45 units
- D60 units
Explanation
AC = 2x + 40 + 1800/x. dAC/dx = 2 - 1800/x² = 0 gives x² = 900, so x = 30. Second derivative 3600/x³ is positive, so it is a minimum. Setting marginal cost 4x + 40 equal to AC gives the same x = 30. Choosing 20 or 45 ignores the fixed-cost term in the derivative.
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