Question

Suppose bicycle pedals cost $20 each and brake pads cost $5 each. If a is the number of pedals you buy and b is the number of brake pads you buy, and you must spend $1000, use the method of Lagrange Multipliers to determine the quantities which maximize and minimize the utility, U=8a²+ab+b², and give the maximum and minimum values of U.

Answer 1

To maximize and minimize the utility function U = 8a² + ab + b² under the budget constraint of spending $1000, we can use the method of Lagrange Multipliers.

Step-by-step explanation:

To maximize and minimize the utility function U = 8a² + ab + b² under the budget constraint of spending $1000, we can use the method of Lagrange Multipliers. Firstly, let's define the Lagrangian function L:

L = 8a² + ab + b² - λ(20a + 5b - 1000)

To find the maximum and minimum values, we need to calculate the partial derivatives of L with respect to a, b, and λ, and set them equal to zero:

∂L/∂a = 16a + b - 20λ = 0

∂L/∂b = a + 2b - 5λ = 0

∂L/∂λ = 20a + 5b - 1000 = 0

Solving these equations simultaneously, we can find the values of a, b, and λ that maximize and minimize U.