24.2. THE METHOD OF LAGRANGE MULTIPLIERS 513

24.2 The Method of Lagrange MultipliersAs an application of the implicit function theorem, consider the method of Lagrange mul-tipliers. Recall the problem is to maximize or minimize a function subject to equalityconstraints. Let f : U → R be a C1 function where U ⊆ Rn and let

gi (x) = 0, i = 1, · · · ,m (24.18)

be a collection of equality constraints with m < n. Now consider the system of nonlinearequations

f (x) = a

gi (x) = 0, i = 1, · · · ,m.

Recall x0 is a local maximum if f (x0) ≥ f (x) for all x near x0 which also satisfies theconstraints 24.18. A local minimum is defined similarly. Let F : U ×R→ Rm+1 be definedby

F (x,a)≡

f (x)−ag1 (x)

...gm (x)

 . (24.19)

Now consider the m+1×n matrixfx1 (x0) · · · fxn (x0)

g1x1 (x0) · · · g1xn (x0)...

...gmx1 (x0) · · · gmxn (x0)

 .

If this matrix has rank m+1 then some m+1×m+1 submatrix has nonzero determinant.See Theorem 20.2.15. It follows from the implicit function theorem, there exists m+ 1variables xi1 , · · · ,xim+1 such that the system

F (x,a) = 0 (24.20)

specifies these m+ 1 variables as a function of the remaining n− (m+1) variables and ain an open set of Rn−m. Thus there is a solution (x,a) to 24.20 for some x close to x0whenever a is in some open interval. Therefore, x0 cannot be either a local minimum ora local maximum. It follows that if x0 is either a local maximum or a local minimum,then the above matrix must have rank less than m+1. It follows that some row is a linearcombination of the others. Thus there exist m scalars,

λ 1, · · · ,λ m,

and a scalar µ , not all zero such that

µ

 fx1 (x0)...

fxn (x0)

= λ 1

 g1x1 (x0)...

g1xn (x0)

+ · · ·+λ m

 gmx1 (x0)...

gmxn (x0)

 . (24.21)