Question

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f(x,y,z) = xyz subject to x² + 2y² + 4z² = 9.

Answer 1

To find the maximum and minimum values of the function f(x,y,z) = xyz subject to the given constraint x² + 2y² + 4z² = 9, we can use Lagrange multipliers.

Step-by-step explanation:

To find the maximum and minimum values of the function f(x,y,z) = xyz subject to the constraint x² + 2y² + 4z² = 9, we can use Lagrange multipliers. First, we define the Lagrangian function as L(x,y,z,λ) = xyz - λ(x² + 2y² + 4z² - 9). To find the critical points, we take the partial derivatives of L with respect to x, y, z, and λ, and set them equal to zero. Solving this system of equations will give us the critical values, which we can then substitute back into the original function to find the maximum and minimum values.