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Question: Suppose a manufacturer's production function is given by: 16q = 65 – 4(1- 4)² – 2(k – 5)² And the cost to the manufacturer is $8 per unit of labor and $16 per unit of capital, so that the total cost (in dollars) is 81 + 16k. The selling price of the product is $64. i. Express the profit as a function of I and k. Give your answer in expanded form. ii. Find all critical points of the function obtained in part (i). Apply the second-order derivative test at each critical point. If the profit is a relative maximum at a critical point, compute the corresponding relative maximum profit. ii. The profit may be considered as a function of I, k, and q, subject to the constraint:

Question: Suppose a manufacturer's production function is given by: 16q = 65 – 4(1- 4)² – 2(k – 5)² And the cost to the manufacturer is $8 per unit of labor and $16 per unit of capital, so that the total cost (in dollars) is 81 + 16k. The selling price of the product is $64. i. Express the profit as a function of I and k. Give your answer in expanded form. ii. Find all critical points of the function obtained in part (i). Apply the second-order derivative test at each critical point. If the profit is a relative maximum at a critical point, compute the corresponding relative maximum profit. ii. The profit may be considered as a function of I, k, and q, subject to the constraint:

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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1 is a and that of good 2 is b'". Question: Suppose a manufacturer's production function is given by: 16q = 65 – 4(1 – 4)² – 2(k – 5)2 And the cost to the manufacturer is $8 per unit of labor and $16 per unit of capital, so that the total cost (in dollars) is 81 + 16k. The selling price of the product is $64. i. Express the profit as a function of I and k. Give your answer in expanded form. ii. Find all critical points of the function obtained in part (i). Apply the second-order derivative test at each critical point. If the profit is a relative maximum at a critical point, compute the corresponding relative maximum profit. iii. The profit may be considered as a function of I, k, and q, subject to the constraint: 16q = 65 – 4(1 – 4)? – 2(k – 5)2 Use the method of Lagrange multipliers to find all critical points of the profit function, subject to the constraint. -4- Final Page
Transcribed Image Text:1 is a and that of good 2 is b'". Question: Suppose a manufacturer's production function is given by: 16q = 65 – 4(1 – 4)² – 2(k – 5)2 And the cost to the manufacturer is $8 per unit of labor and $16 per unit of capital, so that the total cost (in dollars) is 81 + 16k. The selling price of the product is $64. i. Express the profit as a function of I and k. Give your answer in expanded form. ii. Find all critical points of the function obtained in part (i). Apply the second-order derivative test at each critical point. If the profit is a relative maximum at a critical point, compute the corresponding relative maximum profit. iii. The profit may be considered as a function of I, k, and q, subject to the constraint: 16q = 65 – 4(1 – 4)? – 2(k – 5)2 Use the method of Lagrange multipliers to find all critical points of the profit function, subject to the constraint. -4- Final Page
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