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CMA Foundation · Fundamentals of Business Mathematics and Statistics · Calculus - Application in Business

A Kolkata firm has total cost C(x) = x³/3 − 4x² + 30x + 200 (in rupees) where x is output in units. At which output is the marginal cost minimum, and what is that minimum marginal cost?

Marginal cost is minimum at x = 4 with a value of ₹14. Marginal cost is x² − 8x + 30; setting its derivative 2x − 8 to zero gives x = 4, and substituting gives 16 − 32 + 30 = 14, with a positive second derivative confirming the minimum.

  1. Ax = 4, MC = 14Correct
  2. Bx = 8, MC = 30
  3. Cx = 4, MC = 30
  4. Dx = 5, MC = 5

Explanation

MC = x² − 8x + 30. Its derivative 2x − 8 = 0 gives x = 4, and the second derivative 2 is positive, so it is a minimum. MC at 4 = 16 − 32 + 30 = 14. The value 30 is the MC at x = 0 or 8, not the minimum.

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