As an application of the implicit function theorem, consider the method of Lagrange
multipliers from calculus. Recall the problem is to maximize or minimize a function
subject to equality constraints. Let f : U → ℝ be a C1 function where U ⊆ ℝn and
let
gi(x) = 0, i = 1,⋅⋅⋅,m (6.15)
(6.15)
be a collection of equality constraints with m < n. Now consider the system of nonlinear
equations
f (x) = a
g(x) = 0, i = 1,⋅⋅⋅,m.
i
x0 is a local maximum if f
(x0)
≥ f
(x)
for all x near x0 which also satisfies the
constraints 6.15. A local minimum is defined similarly. Let F : U ×ℝ → ℝm+1 be defined
by
( f (x)− a )
| g1(x) |
F (x,a) ≡ || . || . (6.16)
( .. )
gm (x )
(6.16)
Now consider the m + 1 × n Jacobian matrix, the matrix of the linear transformation,
D1F
If this matrix has rank m + 1 then some m + 1 ×m + 1 submatrix has nonzero determinant. It
follows from the implicit function theorem that there exist m + 1 variables, xi1,
⋅⋅⋅
,xim+1 such
that the system
F (x,a) = 0 (6.17)
(6.17)
specifies these m + 1 variables as a function of the remaining n−
(m + 1)
variables and a in
an open set of ℝn−m. Thus there is a solution
(x,a)
to 6.17 for some x close to x0 whenever a
is in some open interval. Therefore, x0 cannot be either a local minimum or a 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 which requires the rows to be linearly dependent. Thus, there
exist m scalars,
are linearly independent, then, μ≠0 and dividing by μ yields an expression of the
form
( ) ( ) ( )
fx1 (x0) g1x1 (x0) gmx1 (x0)
|( ... |) = λ1|( ... |) + ⋅⋅⋅+ λm |( ... |) (6.20)
fx (x0) g1x (x0) gmx (x0)
n n n
(6.20)
at every point x0 which is either a local maximum or a local minimum. This proves the
following theorem.
Theorem 6.5.1Let U be an open subset of ℝnand let f : U → ℝ be a C1function. Then if x0∈ U is either a local maximum or local minimum of f subject tothe constraints 6.15, then 6.18must hold for some scalars μ,λ1,
⋅⋅⋅
,λmnot all equal tozero. If the vectors in 6.19are linearly independent, it follows that an equation of theform 6.20holds.