What Lagrange multipliers are, and what they are good for.

Lagrange multipliers method is a cool process for finding extrema on a bounded surface.

Given a function f, local extrema can be found by finding where the gradient of f is zero. When f is bounded, extrema may lie along this boundary, and these can not be found with the gradient function. One way to find these extrema is by parameterizing the boundary, but this can be very messy in many situations. Another method for finding these extrema is the Lagrange multiplier method, which you will be exploring today.

The idea behind this method is that at the extrema along the boundary, the gradients of f and g will be parallel to each other. One can see this concept with this picture:

(image from Williamson & Trotter, Multivariable Mathematics, second edition, Prentice Hall, p 273)

This can also be generalized for a surface f with many boundaries in many dimensions with the following equation:

Next, one must solve a system of equations consisting of the coordinates of the above function and each of the boundaries.

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