Question

Using Lagrange multipliers, find the global maximum and minimum values of the function f(x, y) = x² + y² - 4y subject to the constraint x² + y² = 9.

Answer 1

To find the global maximum and minimum of the function f(x, y) = x² + y² - 4y given the constraint x² + y² = 9, we use Lagrange multipliers by introducing a new function L(x, y, λ), setting its derivatives to zero, and solving the resulting system of equations.

Step-by-step explanation:

Using Lagrange multipliers, we want to find the global maximum and minimum values of the function f(x, y) = x² + y² - 4y subject to the constraint x² + y² = 9. To do this, we introduce a Lagrange multiplier λ and consider the function L(x, y, λ) = f(x, y) - λ(g(x, y) - c), where g(x, y) = x² + y² is our constraint function, and c = 9 is the constant value it must equal.

The first step is to find the partial derivatives of L with respect to x, y, and λ, set them to zero, and solve the system of equations:

• 
  
• ∂L/∂x = 2x - 2λx = 0
• 
  
• ∂L/∂y = 2y - 4 - 2λy = 0
• 
  
• ∂L/∂λ = x² + y² - 9 = 0

We solve this system to find the points (x, y) that satisfy both the function's condition and the constraint. After solving, we may find more than one point; each must be tested in the original function f(x, y) to determine which yields the maximum and minimum values.