Final answer:
Lagrange multipliers are utilized to find the extremal values of a function subject to a constraint by equating the gradient of the function to a scalar multiple of the gradient of the constraint. This method leads to a system of equations that provides critical points, which are then tested and evaluated to determine the maximum and minimum values of the function.
Step-by-step explanation:
The problem here involves using Lagrange multipliers to find the maximum and minimum values of the function f(x, y, z) = x^2 − 18y + 20z^2 subject to the constraint x^2 + y^2 + z^2 = 1. To apply Lagrange multipliers, we introduce a new variable λ (lambda), which represents the multiplier, and we consider the gradients of the objective function f and the constraint function g(x, y, z) = x^2 + y^2 + z^2 - 1. Our goal is to find the points where ∇f = λ∇g. This leads us to a system of equations derived from the components of the gradients. We then solve this system to find the critical points that may correspond to the maximum and minimum values of f given the constraint. For each critical point identified, we test whether it satisfies the constraint, and we evaluate f at these points to determine the maximum and minimum values.