Applications of Differentiation in Economics [Maxima & Minima]

Applications of Differentiation in Economics

[Maxima & Minima]

1. The total cost C (x) associated with producing and marketing x units of an item is given by \(\mbox{C}\; \left( x \right)\; =\; 0.005x^{3}\; -\; 0.02x^{2}\; +\; 30x\; +\; 3000\), Find

a) Total cost when output is 4 units.

b) Average cost of output of 10 units.

c) Marginal cost when output is 3 units.

2. Find (i) the average total cost (ii) the marginal cost function for each of the following total cost functions. Evaluate them at x = 3 and x = 5.

a) \(T\mbox{C}\; =\; 3x^{2}\; +\; 7x\; +\; 12\)

b) \(T\mbox{C}\; =\; 35\; +\; 5x\; -\; 2x^{2}\; +\; 2x^{3}\)

c) \(T\mbox{C}\; =\; 2x^{2}\; +\; 5x\; +\; 7\)

d) \(T\mbox{C}\; =\; 33\; +\; 2x\; -\; x^{2}\; +\; 4x^{3}\)

3. Find (i) average profit function (ii) marginal profit function for the profit function \(\pi \; =\; x^{2}\; -\; 13x\; +\; 78\)for x = 3 and x = 5.

4. Find (i) average profit function (ii) marginal profit function for the profit function \(\pi \; =\; x^{2}\; -\; 7x\; +\; 64\) for x = 4 and x = 5.

5. Find (i) average revenue function and, (ii) marginal revenue function for the following total revenue function 

a) \(TR\; =\; 12x\; -\; x^{2}\)

b) \(TR\; =\; 27x\; -\; \frac{x^{2}}{3}\; +\; x^{3}\)

Evaluate them at x = 3 and x = 5.

6. Find Marginal Cost (MC) function from the following average cost AC function \(A\mbox{C}\; =\; \frac{160}{x}\; +\; 5\; +\; 3x\; +\; 2x^{2}\).

7. Find the maximum or the minimum values or extreme values of \(2x^{3}\; -\; 9x^{2}\; +\; 12x\; +6\).

8. What will be the values of x for the maximum and minimum values of \(x^{3}\; -\; 9x^{2}\; +\; 24x\; -\; 12\)?

9. Find the maximum and the minimum values of

a) \(2x^{3}\; -\; 21x^{2}\; +\; 36x\; -\; 20\)

b) \(2x^{3}\; -\; 9x^{2}\; +\; 12x\; -1\)

c) \(x^{3}\; +\; 2x^{2}\; -\; 4x\; -\; 8\)

d) \(2x^{2}\; -\; x^{3}\)

e) \(x^{2}\; -\; 3x\; +\; 1\)

f) \(2x^{2}\; +\; 3x^{2}\; -\; 36x\; +\; 10\)

10. Show that the maximum values of \(x\; +\; \frac{1}{x}\) is less than its minimum value.

11. Find for which values of x the following function is a maximum or a minimum :

\(4x^{3}\; -\; 15x^{2}\; +\; 12x\; -\; 2\)

12. A steel produce x tons of steel per week at a total cost of Rs.\(\frac{1}{3}x^{3}\; -\; 5x^{2}\; +\; 99x\; +\; 35\) . find the output level at which the marginal cost attains its minimum.

13. Suppose a manufacturer can sell x items per week at a price \(P\; =\; 20\; -\; 0.001x\) rupees each when it costs \(TC\; =\; 5x\; +\; 2000\).   rupees to produce x items. Determine the number of items he should produce per week for maximum profit.

14. A plant produces x tons of steel per week at a total cost of  Rs.\(\frac{x^{3}}{3}-\; 7x^{2}\; +\; 111x\; +\; 50\) . Find the output level at which the marginal cost attains its minimum.

15. The market demand function of a firm is given by \(4p\; +\; x\; -\; 16\; =\; 0\) and the average cost function takes the form : \(A\mbox{C}\; =\; \frac{4}{x}\; +\; 2\; -\; 0.3x\; +\; 0.05x^{2}\)  where p and x denote the price and the quantity respectively. Find x which gives the maximum profit. Verify also the second order condition.

16. Find the values of x for which the following function is a maximum or a minimum \(y\; =\; \frac{2x\; -\; 1}{x^{2}\; -\; 8x\; -2}\).

17. Determine the critical values of the following function. Investigate the relative maximum or minimum at these critical values \(f\left( x \right)\; =\; 3x^{3}\; -\; 36x^{2}\; +\; 135x\; -13\).

18. Find the maximum balance of the following total revenue (TR) and total profit (\(\pi \)) functions :

a) \(TR\; =\; 32x\; -\; x^{2}\)

b) \(TR\; =\; -x^{2}\; +\; 11x\; -\; 24\)

19. Prove that marginal cost (MC) must equal marginal revenue (MR) at the profit maximising level of output.

20. \(TR\; =\; 1400\; x\; -\; 7.5\; x^{2}\; ,\; \; T\mbox{C}\; =\; x^{3}\; -\; 6x^{2}\; +\; 140x\; +\; 750\). Use MR = MC condition to (a) maximise profit and (b) check the second order condition.

21. The total cost function for producing x units of a commodity is \(T\mbox{C}\; =\; 60\; -\; 12x\; +\; 2x^{2}\) . Find the level of output at which TC is minimum.

22. The demand function faced by a firm is \(P\; =\; 500\; -\; 0.2x\) and its cost function is \(\mbox{C}\; =\; 25x\; +\; 10,000\) where P is price, x is output  and C is cost. Find the output at which the profit of the firm is maximum. What will be the equilibrium output and profit if the government gives a lump sum subsidy of 1000.

23. From each of the following Total Cost (TC) function : 

\(T\mbox{C}\; =\; x^{3}\; -\; 5x^{2}\; +\; 60x\) and \(T\mbox{C}\; =\; x^{3}\; -\; 21x^{2}\; +\; 500x\)

find :

a) The average cost (AC) function.

b) The critical value at which AC is minimised and

c) The minimum average cost.

24. A firm’s demand function is \(3p\; +\; x\; =\; 48\; \). Find the level of output where total revenue is maximised. Also if the average cost function is given by \(A\mbox{C}\; =\; x^{2}\; -\; 2x\; +\; 10\) , find the level of output which minimises marginal cost.

25. A firm has the following total revenue and total cost functions \(TR\; =\; 100x\; -\; x^{2}\) , \(T\mbox{C}\; =\; x^{3}\; -\; \frac{57}{2}x^{2}\) , where x is the level of output. Determine the maximum profit.

26. A monopolist produces x sets per day at the total cost of \(\frac{x^{2}}{25}\; +\; 3x\; +\; 100\) . Show that if the demand curve is \(x\; =\; 75\; -\; 3p\) , where p is the price per set, he will produce about 30 sets. Find the monopoly price.

27. The marginal cost of an item consists of Rs. 1200 as overhead cost, material cost of Rs. 4 per item and the labour cost of Rs.\(\frac{3x^{2}}{256}\) for x items produced. Find how many items be produced to have the average cost as minimum.

28. If a monopoly has a total cost of \(T\mbox{C}\; =\; ax^{2}\; +\; bx\; +\; c\) and if the demand law is \(p\; =\; \beta \; -\; \alpha x^{2}\), then show that the output for maximum profit is \(x\; =\; \frac{\sqrt{a^{2}\; +\; 3\alpha \left( \beta -b \right)\; }\; -\; a}{3\alpha }\).

29. The demand function is \(x\; =\; 100\; -\; 20\; \sqrt{p}\) and \(T\mbox{C}\; =\; \frac{1}{25}x^{2}\; +\; 3x\; +\; 100\) . Show that for maximum revenue the producer should produce 25 sets per week. Also find the monopoly price.

30. Given the total cost function \(T\mbox{C}\; =\; ax^{2}\; +\; bx\; +\; c\) and the demand function \(p\; =\; \alpha \; +\; \beta x\) . Find the  output when the monopolist (i) Fix the price (ii) fix the demand.

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