Question

Minimize 2x^2 + y^2 ​, subject to the constraint 33 - x - 4y = 0 Question content area bottom Part 1 The minimum value of the function is enter your response here. ​(Type an exact answer in simplified​ form.)

Answer 1

Lagrange multipliers minimize
`2x^2 + y^2`subject to 33 - x - 4y = 0. The minimum is 266 at (x,y) = (11,5).

This problem can be solved using the method of Lagrange multipliers. Here's how:

Set up the Lagrangian: Define a new function called the Lagrangian, which is the original function plus a Lagrange multiplier λ multiplied by the constraint equation. In this case, the Lagrangian L(x, y, λ) =
`2x^2 + y^2`+ λ(33 - x - 4y).

Find the partial derivatives: Take the partial derivatives of L with respect to x, y, and λ, and set them equal to zero. This gives you a system of three equations:

∂L/∂x = 4x - λ = 0

∂L/∂y = 2y - 4λ = 0

∂L/∂λ = 33 - x - 4y = 0

Solve the system of equations: Solve the system of equations for x, y, and λ. You can do this by manipulating the equations to eliminate λ and then solving for x and y. In this case, you'll find:

x = 11

y = 5

Evaluate the original function: Substitute these values of x and y back into the original function to find the minimum value. In this case, the minimum value is:

`2(11)^2 + 5^2`= 266

Therefore, the minimum value of the function is 266, which is achieved at the point (x, y) = (11, 5).