CA Foundation · Quantitative Aptitude · Differential and Integral Calculus
A firm's total cost function is C(x) = x² + 3600, where x is the number of units produced. The output at which the average cost per unit is minimum is:
Average cost is minimum at 60 units. Dividing cost by x gives AC = x + 3600/x. Setting its derivative 1 − 3600/x² to zero gives x = 60, and the positive second derivative confirms a minimum. The figure 120 is the minimum average cost, not the output.
- A30 units
- B60 unitsCorrect
- C120 units
- D3600 units
Explanation
Average cost AC = C/x = x + 3600/x. Setting dAC/dx = 1 − 3600/x² = 0 gives x² = 3600, so x = 60. The second derivative 7200/x³ is positive there, confirming a minimum. The value 120 is the minimum AC itself, not the output.
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