Can you explain how to use the method of Lagrange multipliers to find the maximum and minimum values of a function subject to a constraint?

The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. Suppose we have a function f(x, y, …) that we want to maximize or minimize subject to a constraint g(x, y, …) = 0. The method involves introducing a new variable, λ (the Lagrange multiplier), and studying the Lagrange function L(x, y, …, λ) = f(x, y, …) – λ(g(x, y, …) – c). We then find the stationary points of L by solving the system of equations given by the partial derivatives of L with respect to all variables (including λ) being equal to zero. These points give the candidates for the extrema of f subject to the constraint g.

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