Question

Use the Lagrange Multiplier Theorem to find the global minimum of x₁ + 2x₂ subject to x₁² + x₂² = 5.

Answer 1

To find the global minimum using the Lagrange Multiplier Theorem, set up equations, solve them simultaneously, and substitute the values back into the objective function.

Step-by-step explanation:

To find the global minimum of the function x₁ + 2x₂ subject to the constraint x₁² + x₂² = 5 using the Lagrange Multiplier Theorem, we can set up the following equations:

L = x₁ + 2x₂ - λ(x₁² + x₂² - 5) and

∂L/∂x₁ = 1 - 2λx₁ = 0,

∂L/∂x₂ = 2 - 2λx₂ = 0,

∂L/∂λ = x₁² + x₂² - 5 = 0.

Solving these equations simultaneously, we can find the values of x₁, x₂, and λ. Substituting these values back into the objective function x₁ + 2x₂ will give us the minimum value.