Business math for hons part 02

Application for diff. calculus

1. Find the inflection point of the function Y = x3-3x2+3x
2. Show that the function Y = xe-x has a point of inflection at x = 2
3. The difference of two numbers is 100. The square of the larger one exceeds five times the square of the smaller one by an amount which is maximum, Find the numbers.
4. Prove that the curve given by 3y = x3 – 3x2 – 9x + 11, has a maximum at x = -1, minimum at x= 3, and point of inflexion when x=1
5. The cost C of manufacturing a certain article is given by the formula C= 5+48\x + 3x2 where x the number of articles manufactured. Find the minimum value of C.
6. A company finds that it can sell out a certain product that it produces, at the rate of tk 2 per 
Unit. It estimates the cost function of the product to be tk. [1000+1/2(q/50)2] for q units produced. (I) find the expression for the total profit, if q units are produced and sold.
(II) Find the number of units produced that will maximize profit. (III) What is the amount of this maximum profit? (IV) What would be the profit if 6000 units are produced?
7. A sitar manufacturer notices that he can sell x sitars per week at p taka each where 5x= 375- 3p. The cost of production is (500+ 13x+ x2/5) taka. Show that the maximum profit is obtained when the production is 30 sitars per week.
8. If the demand function of the monopolist is 3q= 98-4p and average cost is 3q+2, where q is output and p is the price, find maximum profit of the monopolist.
9. The cost function of a firm is C= 300x - 10x2 + 1/3 x3, where C stands for cost and x for output. Calculate (i) output a marginal cost is minimum. (ii) Output at which average cost is minimum. (iii) Output at which average cost is equal to marginal cost.
10. Find the profit maximizing output given the following revenue and cost functions: R(q) = 1000q-2q2, C(q)= q3-59q2+1315q+2000. Also find the maximum profit. 
11. The total revue function of a firm is given as R=21q-q2 and its total cost function as C = q3/3 - 3q2-7q+16, where q is the output. Find (i) the output at which the total revenue is maximum and minimum.
12. The demand function of a consumer good is x= 1/3(25-2p), where x is the number of units and p is the price. The average cost per unit is tk 40. Find: (a) the revenue function R in terms of price P; (b) the cost function C (c) the profit function P, (d) the price per unit that maximizes the profit function, and (e) the maximum profit.
13. The demand function faced by a firm is P= 500-0.2x and its cost function is C=25x+10000 where p=price, x=output and C=cost. Find the output at which the profit of the firm is maximum; also find the price it will charge.
14. The production function of a consumer product is given by Q= 40x + 3x2 – x3/3, where Q is the total output and x is the units of input. (i) Find the number of units of input required to give maximum output. (ii) Find the maximum value of marginal product. (iii) Verify that when the average product is maximum, it is equal to marginal product.
15. The total cost function of a firm is C=x3/3 – 5x2 +28x +10, where C is total cost and x is output. A tax at the rate of tk 2 per unite of output is imposed and the producer adds it to his cost. If the market demand function is given by p=2530-5x, where tk p is the price per unit of output, find the profit maximizing output and price.