Use Lagrange multipliers to solve                                                           Maximize f(x, y, z) = xyz Constraints: x + y + z = 32,   x − y + z = 0

Use Lagrange multipliers to find the maximum and minimum values of f(x, y, z) = x + 2y + 5z, subject to the constraint x² + y² + z² = 9, if such values exist. maximum minimum = (For either value, enter DNE if there is no such value.)

Use Lagrange multipliers to find the maximum and minimum values of f(x, y) = 4x − y subject to the constraint x² + 2y² = 66, if such values exist. maximum = minimum = (For either value, enter DNE if there is no such value.)

Minimize the function f(x, y, z) = x² + y? + z² subject to the constraint 6x + 12y + 12z = 54. Function value at the constrained minimum: