Lagrange Function and Partial Derivatives

The video tutorial explains how to solve an optimization problem using Lagrange's method. It begins with a problem statement involving a utility function and budget constraint. The instructor sets up the Lagrange function and takes partial derivatives to form equations. By solving these equations, the optimal quantities of goods are determined, maximizing the consumer's utility within the given budget.

What is the utility function given in the problem?

• U(X, Y) = X^2 * Y^2

• U(X, Y) = X^(3/4) * Y^(3/4)

• U(X, Y) = X^(1/2) * Y^(1/2)

• U(X, Y) = X^3 + Y^3

What are the prices of goods X and Y?

• X: $6, Y: $3

• X: $5, Y: $4

• X: $3, Y: $6

• X: $4, Y: $6

What is the consumer's income in the problem?

• $120

• $200

• $100

• $150

What is the budget constraint equation?

• 6X + 3Y = 100

• 6X + 3Y = 150

• 6X + 3Y = 120

• 6X + 3Y = 200

What is the first step in using Lagrange's method?

• Differentiate the utility function

• Multiply the utility function by Lambda

• Set the utility function to zero

• Set the budget constraint to zero

What is the Lagrange function composed of?

• Utility function only

• Budget constraint only

• Utility function and budget constraint

• None of the above

What do you do after forming the Lagrange function?

• Solve for X and Y

• Take partial derivatives

• Set the Lagrange function to zero

• Multiply by Lambda

What is the partial derivative of the Lagrange function with respect to X?

• 3/4 * X^(3/4) * Y^(3/4) - 6λ

• 3/4 * X^(3/4) * Y^(1/4) - 6λ

• 3/4 * X^(1/4) * Y^(1/4) - 6λ

• 3/4 * X^(1/4) * Y^(3/4) - 6λ

What is the partial derivative of the Lagrange function with respect to Y?

• 3/4 * Y^(1/4) * X^(3/4) - 3λ

• 3/4 * Y^(3/4) * X^(1/4) - 3λ

• 3/4 * Y^(3/4) * X^(3/4) - 3λ

• 3/4 * Y^(1/4) * X^(1/4) - 3λ

What is the result of dividing the first equation by the second equation?

• X = Y

• Y = X

• X = 2Y

• Y = 2X