Question

Explain the Method of Lagrange Multipliers for solving constrained optimization problems.

Short answer

The method of Lagrange multipliers is a technique for finding the extreme values (maxima, minima or saddle points) of a function subject to equality constraints. It incorporates the constraint into the function itself through the Lagrange function, and solutions are obtained by finding the points where the gradients of the function and the constraint are parallel.

Step-by-step solution

Understanding the Method

Lagrange method is used to find the local maxima and minima of a function subject to equality constraints. It introduces a new variable (λ, the Lagrange multiplier) for each constraint. It relies on the principle that at the maximum and minimum, the function's gradient is either zero or parallel to the constraint's gradient.

Formulating the Lagrange Function

Given an optimization problem of a function f(x, y) with a constraint g(x, y) = 0, we create a new function, the Lagrange function L(x, y, λ) = f(x, y) - λ*g(x, y). The new variable λ is the Lagrange multiplier.

Solving the Lagrange Function

We solve for the points (x, y, λ) that make the partial derivatives of the Lagrange function with respect to all variables equal to zero, yielding a system of equations. i.e \(\frac{\partial L}{\partial x} = 0\), \(\frac{\partial L}{\partial y} = 0\) and \(\frac{\partial L}{\partial \lambda} = 0\).

Maxima and Minima

By solving this system of equations, we find the points that satisfy both the original function and the constraint. At these points, the function has a local maximum or minimum under the constraint. Determine if these points are local maxima, minima, or saddle points using the second derivative test.