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CA Foundation · Quantitative Aptitude · Differential and Integral Calculus

A firm has demand p = 100 − 0.5x and total cost C(x) = 20x + 200, where p is in ₹ and x is output. A tax of ₹10 per unit is imposed on the firm, increasing its total cost by 10x. What is the maximum profit after the tax?

After the tax, cost is 30x + 200, so profit is −0.5x² + 70x − 200. Maximum is at x = 70, giving 2,250 rupees. None of the listed values equals this, so this question is invalid.

  1. A₹1,800Correct
  2. B₹2,000
  3. C₹1,600
  4. D₹2,200

Explanation

After tax, cost = 30x + 200. Profit = (100x − 0.5x²) − 30x − 200 = −0.5x² + 70x − 200. P' = −x + 70 = 0 gives x = 70. P = −0.5(4900) + 4900 − 200 = −2450 + 4900 − 200 = ₹2,250? Recheck: 70x = 4900, so P = 4900 − 2450 − 200 = 2250.

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