4.12. EXERCISES 109

13. One of the big applications of the implicit function theorem is to the method ofLagrange multipliers. The heuristic explanations usually given in beginning calculuscourses are specious. At least this is certainly true of the explanation I use all thetime based on pictures and geometric reasoning. They break down as soon as youask the obvious question whether there is a smooth curve through a point in the levelsurface. In other words, why does the level surface even look the way we draw it inthese courses? To do the method of Lagrange multipliers correctly, you need to usesome sort of big theorem and the version involving the implicit function theorem islikely the easiest. Using the implicit function theorem, prove the following theoremwhich is the general method of Lagrange multipliers.

Theorem 4.12.1 Let U be an open subset of Rn and let f : U → R be a C1

function. Then if x0 ∈U, has the property that

gi (x0) = 0, i = 1, · · · ,m, gi a C1function, and x0 is either a local maximum or localminimum of f on the intersection of the level sets {x : gi (x) = 0} i = 1, · · · ,m, and ifsome m×m submatrix of

Dg(x0)≡

 g1x1 (x0) g1x2 (x0) · · · g1xn (x0)...

......

gmx1 (x0) gmx2 (x0) · · · gmxn (x0)

has nonzero determinant, then there exist scalars, λ 1, · · · ,λ m such that fx1 (x0)

...fxn (x0)

= λ 1

 g1x1 (x0)...

g1xn (x0)

+ · · ·+λ m

 gmx1 (x0)...

gmxn (x0)

 (4.26)

holds.

Hint: Let F : U×R→ Rm+1 be defined by

F(x,a)≡

f (x)−ag1 (x)

...gm (x)

 . (4.27)

and if the condition holds on rank, and 4.26 fails to hold, then from linear algebrayou can use the implicit function theorem to solve for m+ 1 of the x variables interms of the others, a being one of them, these other variables being in an open set.In particular a cannot be a local extremum unless 4.26 holds.

14. Now consider the queston about level surfaces. Suppose you have

S ={

x ∈ Rn+1 : f (x) = c}.

We usually refer to this as a level surface in Rn+1 and we give examples of thingslike ellipsoids and spheres. Then everyone is deceived into thinking they know whatis going on because of the examples. After this deception, and this is indeed what it