Mathematical tools for intermediate economics classes
Iftekher Hossain
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Derivatives and their uses

Section 8

Practise questions

Question 1

Suppose a competitive firm faces the total cost function \(TC(x) = 326 + 5x^{2} + \frac{1}{3}x^{3}\). Suppose the market price is \(2000\).

  • Find total revenue and marginal revenue functions.
  • Find marginal cost function.
  • Write the profit function.
  • Find the critical values at which the profit may be maximized.
  • Check the second-order condition regarding the slope of MC and MR functions.
  1. \(R(x) = P \cdot x = 2000x\), \({R}'(x) = 2000\)
  2. \(MC(x) = 10x + x^{2}\)
  3. \(\pi (x) = 2000x - (326 + 5x^{2} + \frac{1}{3}x^{3})\)
  4. Critical values are: \(-50, 40\)
  5. Slope of the \(MR\) function is \(0\) $$M{C}'(-50) = -90 < 0$$ $$M{C}'(40) = 70 > 0$$ Slope of \(MC\) \(>\) slope of \(MR\) when \(x = 40\). Profit is maximized when \(x = 40\).


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