Final answer:
Using Lagrange multipliers with the function f(x, y) = x²y² and the constraint xy = 1, you'll find that the maximum value of the function occurs when x = y = 1, which results in f(1, 1) = 1.
Step-by-step explanation:
You're looking to use Lagrange multipliers to find the maximum and minimum values of the function f(x, y) = x²y², given the constraint xy = 1. To begin, set up the Lagrange function L(x, y, λ) = x²y² - λ(xy - 1), where λ is the Lagrange multiplier. Now, differentiate L with respect to x, y, and λ and set each derivative equal to zero to find your potential maximum and minimum points.
For ∂L/∂x, the derivative is 2xy² - λ y = 0. Similarly, for ∂L/∂y, the derivative is 2yx² - λ x = 0. Finally, for ∂L/∂λ, you simply get the constraint xy = 1. Solve the system of equations to find the values of x, y, and λ that satisfy all these conditions simultaneously.
In this case, you'll find that x = y = 1 and λ = 2. Plug these values back into the function f(x, y) to find the maximum value since the constraint curve xy = 1 does not allow for a minimum in the region where both x and y are positive. Thus, the maximum value of f(x, y) given the constraint is f(1, 1) = 1.