514 CHAPTER 24. IMPLICIT FUNCTION THEOREM*

If the rank of the matrix  g1x1 (x0) · · · gmx1 (x0)...

...g1xn (x0) · · · gmxn (x0)

 (24.22)

is m, then we can choose µ = 1 because the columns span Rm. Thus there are scalars λ isuch that fx1 (x0)

...fxn (x0)

= λ 1

 g1x1 (x0)...

g1xn (x0)

+ · · ·+λ m

 gmx1 (x0)...

gmxn (x0)

 (24.23)

at every point x0 which is either a local maximum or a local minimum. This proves thefollowing theorem.

Theorem 24.2.1 Let U be an open subset of Rn and let f : U →R be a C1 function.Then if x0 ∈U is either a local maximum or local minimum of f subject to the constraints24.18, then 24.21 must hold for some scalars µ,λ 1, · · · ,λ m not all equal to zero. If therank of the matrix in 24.22 is m, it follows 24.23 holds for some choice of the λ i.