Final answer:
Using Lagrange multipliers, we find that the maximum value of the function is 4 and the minimum value is -4, by evaluating the function at the points (0, 2), (0, -2), (2, 0), and (-2, 0).
Step-by-step explanation:
We are tasked with using Lagrange multipliers to find the maximum and minimum values of the function f(x, y) = y² - x², subject to the constraint x² + y² = 4. To apply Lagrange multipliers, we set up the following system of equations based on the gradients of the function and the constraint:
- ∇f(x, y) = ∇g(x, y) λ
- ∇f(x, y) = −(2x, 2y)
- ∇g(x, y) = (2x, 2y)
- x² + y² = 4
This yields the system:
- −(2x) = 2xλ
- −(2y) = 2yλ
- x² + y² = 4
From the first two equations we get two cases: either x = 0 or λ = −1. If x = 0, then y² = 4, giving us y = ±2. If λ = −1, then y = 0, and x² = 4, giving us x = ±2. Therefore, we evaluate f(x, y) at (0, 2), (0, −2), (2, 0), and (−2, 0). We find that the maximum value of f(x, y) is 4 (at (0, 2) and (0, −2)) and the minimum value is −4 (at (2, 0) and (−2, 0)). If no solutions were found, we would state that the answer DNE (does not exist).