Question

Please solve and I will leave an upvote. Use the method of Lagrange multipliers to find the maximum of f(x, y) = 4x^2 y subject to the constraint 3x - 4y = -10. Round your answer to two decimal places.

Answer 1

Use the method of Lagrange multipliers to find the maximum of the function f(x, y) = 4x^2y, subject to the constraint 3x - 4y = -10. Introduce Lagrange multiplier λ and set up a system of equations with the gradients of the function and the constraint. Solve the system to find the values x and y that maximize the function.

Step-by-step explanation:

To solve for the maximum of the function f(x, y) = 4x2y subject to the constraint 3x - 4y = -10 using the method of Lagrange multipliers, we need to introduce a new variable, λ (lambda), which is the Lagrange multiplier. We set up the system of equations by taking the gradient of the function and the gradient of the constraint set equal to each other multiplied by λ.

• The partial derivative of f with respect to x: 8xy
• The partial derivative of f with respect to y: 4x2
• The partial derivative of the constraint with respect to x: 3
• The partial derivative of the constraint with respect to y: -4

Setting up the system we get:

• 8xy = 3λ
• 4x2 = -4λ
• 3x - 4y = -10

Solving this system will give us the values of x, y, and λ that maximize our function subject to the given constraint.