Final answer:
Use Lagrange multipliers to find the extremum of f(x, y) = 4x²y subject to the constraint 4x² + 2y² = 384 by setting up the Lagrangian, taking its gradient, solving the resulting equations, and substituting back into f(x, y).
Step-by-step explanation:
To find the maximum and minimum values of the function f(x, y) = 4x²y, given the constraint 4x² + 2y² = 384, we use the method of Lagrange multipliers. This involves setting up a new function, L(x, y, λ) = f(x, y) - λ(g(x, y) - c), where g(x, y) = 4x² + 2y², and c is the constant on the right-hand side of the constraint equation.
We then take the gradient of L and set it equal to zero, yielding three equations: ∇2L/∇x = 8xy - 8λx = 0, ∇2L/∇y = 4x² - 4λy = 0, and the original constraint 4x² + 2y² = 384. Solving these equations simultaneously leads to the possible solutions for x, y, and λ. Substituting these values back into the function f(x, y), we can find the maximum and minimum values.