Final answer:
To find the extreme values of f(x, y) = 2xy using Lagrange multipliers, a proper constraint equation, such as g(x, y) = x² + y² - C = 0, is needed to form and solve the system of equations with the gradients of f and g.
Step-by-step explanation:
The student asked how to use Lagrange multipliers to find the maximum and minimum values of the function f(x, y) = 2xy subject to a constraint. However, the constraint provided, x² + y² + 9, does not seem to be an equation because it lacks an equals sign. To proceed with Lagrange multipliers, we should have a constraint such as g(x, y) = x² + y² - C = 0, where C is a constant value. Although the question is incomplete, the general approach is to solve the system of equations formed by the gradient of f(x, y), the gradient of g(x, y), and the constraint equation g(x, y) = 0. The solutions to this system give the points where the function may achieve its maximum and minimum values.