ABSTRACT
To justify the method of Lagrange multipliers we must show that if P = (a, b) is an “interior” point of the level set g(x, y) = 0 at which the function f(x, y) has a local maximum or minimum then there exists a real number λ such that
∇f(a, b) = λ∇g(a, b). We first examine a very simple example. Consider the problem of finding the local maxima and minima of f(x, y) = 2x2 + y2 on the set x + y = 1. From the equation x + y = 1 we see that y = 1 − x and so the function f takes the values
f(x, 1 − x) = 2x2 + (1 − x)2 = 3x2 − 2x + 1 on the set x + y = 1.
