6.1.4 Optimization with an equality constraint: conditions under which a stationary point is a global optimum
max_{x,y} f(x, y) subject to g(x, y) = c
and either g'_{1}(x*, y*) ≠ 0 or g'_{2}(x*, y*) ≠ 0 then there is a number λ* such that (x*, y*) is a stationary point of the Lagrangean L(x, y) = f(x, y) −
λ*(g(x, y) − c)), given λ*.
The fact that (x*, y*) is a stationary point of the Lagrangean does not mean that (x*, y*) maximizes the Lagrangean, given λ*. (The Lagrangean is a function like any other, and we know that a stationary point of an arbitrary function is not necessarily a maximizer of the function. In an exercise you are asked to work through a specific example.)
Suppose, however, that (x*, y*) does in fact maximize L(x, y), given λ*. Then
L(x*, y*) ≥ L(x, y) for all (x, y),
or
f(x*, y*) − λ*(g(x*, y*) − c) ≥ f(x, y) − λ*(g(x, y) − c) for all (x, y).
Now, if (x*, y*) satisfies the constraint then g(x*, y*) = c, so this inequality is equivalent to
f(x*, y*) ≥ f(x, y) − λ*(g(x, y) − c) for all (x, y),
so that
f(x*, y*) ≥ f(x, y) for all (x, y) with g(x, y) = c.
That is, (x*, y*) solves the constrained maximization problem.
In summary,
if (x*, y*) maximizes the Lagrangean, given λ*, and satisfies the constraint, then it solves the problem.Now, we know that any stationary point of a concave function is a maximizer of the function. Thus if the Lagrangean is concave in (x, y), given λ*, and (x*, y*) is a stationary point of the Lagrangean, then (x*, y*) maximizes the Lagrangean, given λ*, and hence if it satisfies the constraint then it solves the problem. Precisely, we have the following result.
 Proposition

Suppose that f and g are differentiable functions of two variables defined on an open convex set S and suppose that there exists a number λ* and an interior point (x*, y*) of S
that is a stationary point of the Lagrangean
L(x, y) = f(x, y) − λ*(g(x, y) − c).That is,
L'_{1}(x*, y*) = f'_{1}(x*, y*) − λ*g'_{1}(x*, y*) = 0 L'_{2}(x*, y*) = f'_{2}(x*, y*) − λ*g'_{2}(x*, y*) = 0.
 Proof
 Given that f and g are differentiable, so is L, so the fact that L is concave means that by a previous result any stationary point of L is a maximizer of L. The argument before the statement of the result establishes that any maximizer of L that satisfies the constraint g(x, y) = c solves the constrained maximization problem.
 Corollary

Suppose that f is a continuously differentiable function of two variables defined on an open convex set S and g is a linear function of two variables defined on this set. Suppose that there exists a number λ* and an
interior point (x*, y*) of S that is a stationary point of the Lagrangean
L(x, y) = f(x, y) − λ*(g(x, y) − c).Suppose further that g(x*, y*) = c. Then
 Example

Consider the problem
max_{x,y} x^{a}y^{b} subject to px + y = m,considered previously. We found that there is a value of λ* such that
(x*, y*) = am (a + b)p , bm a + b