MATH2003J, OPTIMIZATION IN ECONOMICS,
BDIC 2023/2024, SPRING
Problem Sheet 3
Θ Question 1:
Use the method of Lagrange multipliers to find the maximum and minimum of f (x, y) = 5x2 + 5y2 + 1 subject to the constraint xy = 1.
Θ Question 2:
Use the method of Lagrange multipliers to find the maximum and minimum of f (x, y) = x+3y subject to the constraint x2 + y2 = 10.
Question 3:
Use the method of Lagrange multipliers to find the maximum and minimum of f (x, y) = 2x2 + 2y2 + 1 subject to the constraint x2 + xy + y2 = 6.
Θ Question 4:
Use the method of Lagrange multipliers to find the maximum of f (x1 , x2 , x3 ) = 5x1x2x3 subject to the constraint x1 + 2x2 + 3x3 = 24.
Question 5:
Use the method of Lagrange multipliers to find the maximum and minimum of f (x,y, z) = x + 2y + 2z subject to the constraint x2 + y2 + z2 = 9.
Θ Question 6:
Use the method of Lagrange multipliers to find the maximum and minimum of f (w, x,y, z) = w + x + y + z subject to the constraint w2 + x2 + y2 + z2 = 1.
Θ Question 7:
Use the method of Lagrange multipliers to find the maximum and minimum of f (x,y, z) = y subject to the constraints z = x + y and 2x2 + y2 + 2z2 = 8.
Θ Question 8:
Find the minimum of x2 − 2x + 2y2 + z2 + z subject to the constraints x + y + z = 1 and 2x − y − z = 5.
Question 9:
Find the maximum and the minimum of x + 2y subject to the constraint x2 + y2 = c, where c is a positive constant. Explain why the maximum and the minimum are attained.
Θ Question 10:
A company is planning to sell a new product at the price of ¥125 per unit and estimates that if x euro is spent on training staff and y euro is spent on advertising the product, then
y + 300/50y + x + 200/75x
units of the product will be sold. The cost of manufacturing the product is ¥45 per unit. If the company has a total of ¥5,000 to spend on training staff and advertisement, how should this money be allocated to generate the largest possible profit?
+ Note: the profit function is given by
(No. of units) × (price per unit - cost per unit) - total amount spent on training and advertisement