Extrema in Case of Several Variables

CHRISTIAN C. FENSKE

Extrema in Case of Several Variables a

f a v o u r i t e topic of most calculus courses is the calculation of extrema. Every calculus student is confronted w i t h the following:

S t a n d a r d c a l c u l u s r e s u l t . L e t n >-- 2 a n d J C R a n o p e n i n t e r v a l . L e t f : J - - ) ~ be n - 1 t i m e s d i f f e r e n t i a b l e on J a n d n t i m e s d i f f e r e n t i a b l e at s o m e p o i n t a E J. A s s u m e t h a t f ( k ) ( a ) = O f o r k = 1 , . . . , n - 1 b u t f O O ( a ) r O. T h e n there is the f o l l o w i n g a l t e r n a t i v e : (1) E i t h e r n i s even. T h e n f h a s a n i s o l a t e d e x t r e m u m at a, a n d t h a t i s a m a x i m u m i n case f O O ( a ) < 0 a n d a minimum i n c a s e f ( n ) ( a ) > O. (2) Or n i s odd. T h e n f does n o t a t t a i n a local e x t r e m u m at a. When the course proceeds to functions of more than one variable we meet this theorem again--but now only for s e c o n d d e r i v a t i v e s . B u t w h a t a b o u t a f u n c t i o n w i t h t h e first five derivatives vanishing? Of course, there arises the quest i o n o f w h a t p r e c i s e l y w e m e a n b y " v a n i s h i n g " o f an n - t h d e r i v a t i v e , a n d w h a t w e s h o u l d u s e as a s u b s t i t u t e for t h e conditionf(n)(a) being positive or negative. This again depends on how we define differentiability for functions of several variables. In this p a p e r , I first e x p l a i n h o w t h e t h e o r e m w o u l d l o o k f o r a l o w - b r o w a p p r o a c h . T h e n I will d i s c u s s b r i e f l y t h e modifications required for the high-brow approach where higher derivatives are viewed as multilinear forms. O f c o u r s e , at first g l a n c e o n e s u s p e c t s t h a t t h e multivariable case should be well known, and I am pretty sure it is. A l t h o u g h I h a v e l o o k e d i n t o n u m e r o u s c a l c u l u s t e x t s a n d a s k e d at l e a s t a s m a n y c o l l e a g u e s , I h a v e n o t b e e n a b l e to i d e n t i f y a s o u r c e 9 E i t h e r t h e r e is a p r o o f o f t h i s r e s u l t in t h e l i t e r a t u r e , b u t I d i d n o t f i n d it, o r t h e r e s u l t s e e m e d p l a u s i b l e to e v e r y o n e w h o t h o u g h t o f it, b u t w r i t i n g it d o w n w a s n o t w o r t h w h i l e . S o I j u s t p r e s e n t h e r e a p r o o f for r e f e r e n c e p u r p o s e s a n d m a y b e f o r u s e in c a l c u l u s c o u r s e s .

The L o w - B r o w Approach In t h e l o w - b r o w a p p r o a c h a f u n c t i o n f : U--~ ~ o n an o p e n s e t U C ~,r~ is s a i d to b e n t i m e s c o n t i n u o u s l y d i f f e r e n t i a b l e if all partial d e r i v a t i v e s u p to o r d e r n e x i s t a n d a r e c o n t i n u o u s o n U. (I w r i t e Di for t h e p a r t i a l d e r i v a t i v e w i t h r e s p e c t to t h e i-th variable.) S c h w a r z ' s t h e o r e m o n t h e i n t e r c h a n g e ability o f partial d e r i v a t i v e s t h e n tells u s t h a t for a ~ U a n d h O), . . . , h (~) E ~ ' , t h e m a p d~)~(a): ~m x 9 9 9 x ~m____> w i t h d'~f(a)(h (1), h (ro) = ~ DL . 9 9 Dj f(a)h(~.Jl1) " " 9 h(~ 0 ?~ 9

"

"

~

9

1

( s u m m a t i o n o v e r all d i s t i n c t n - t u p l e s ( j l , 9 9 9 , J~,) w i t h 1 --< j~ --< m ) is a s y m m e t r i c n - l i n e a r m a p . If h ~ ~ m w e w r i t e d ~ f ( a ) ( h n) : = d~J[a)(h, . . . , h). W e t h e n h a v e THEOREM. L e t U be o p e n i n ~.,r, a n d let f : U ~ ~ be n t i m e s c o n t i n u o u s l y d i f f e r e n t i a b l e . L e t 2 <--p <- n, a n d a s s u m e t h a t f o r s o m e a ~ U a n d all h ~ ~m, w e h a v e d f ( a ) ( h ) = d 2 f ( a ) ( h 2) . . . . . d p - l f ( a ) ( h p 1) = 0, b u t dPf(a)(h p) r 0 f o r s o m e h @ ~m. T h e n the f o l l o w i n g holds: (1) A n e c e s s a r y c o n d i t i o n f o r f at a i s t h a t p be even. L e t t h e n p be even. (2) A n e c e s s a r y c o n d i t i o n f o r mum (minimum) at a i s f o r all h E ~,m. (3) A s u f f i c i e n t c o n d i t i o n f o r mum (minimum) at a i s f o r all h E ~m\{0}.

to h a v e a local e x t r e m u m

f to h a v e a local m a x i t h a t d P f ( a ) ( h p) <-- 0 (>--0) f to h a v e a local m a x i t h a t d P f ( a ) ( h p) < 0 ( > 0 )

P r o o f T o b e g i n with, c h o o s e y o u r f a v o u r i t e n o r m I1"11o n ~m. (1) L e t p b e o d d . W e h a v e t o s h o w t h a t f d o e s not have a l o c a l e x t r e m u m at a. B y a s s u m p t i o n t h e r e is a n h s u c h t h a t d P f ( a ) ( h p) #= O. U p o n d i v i d i n g h b y its n o r m w e m a y a s s u m e t h a t Ilhll = 1. S i n c e p is o d d w e m a y

@ 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 1, 2003

49

e v e n a s s u m e t h a t / ~ :-- dPf(a)(h ; ) < 0 (else w e w o u l d r e p l a c e h b y - h ) . N o w w e s e t 9 := -t~/p! an d use T a y l o r ' s f o r m u l a ([J, C o r o l l a r y 8.17]) to find a 5 > 0 such that

entiable at a E D if t h e r e is a linear m ap T E L(E, F) and there is a function r : D --~ F c o n t i n u o u s at a, s u c h that r(a) = 0 a n d f ( x ) - f ( a ) + T ( x - a) + I]x - aNr(x ) for all x E D.

[f(a + k) - f ( a )

- ~.dPf(a)(kP)] ~ 2[Ikl~'

w h e n e v e r k ~ ~m satisfies Ilk]] < & Let 0 < t < 8 . We t h e n h a v e f ( a + t h ) - f ( a ) < ~ t P + • p = ~ t , < 0. So t h e r e c a n n o t be a l o cal minp~ 2p~ 2 1mum at a. B u t w e m a y as well c h o o s e an h w i t h ]]h[I = 1 s u c h t h a t tL' = dPf(a)(h p) > O. Let t h e n 9 :=tz'/p!. We c h o o s e ~ a n d t as a b o v e a n d find that

0 < I~' tp = -e--tP + ~'tP 2p! 2 2p! <--f(a + th) - f ( a ) . So w e d o n ' t h a v e a lo c a l m a x i m u m at a either. (2) N o w a s s u m e t h a t w e h a v e a l o c a l m i n i m u m at a (else w e r e p l a c e f w i t h -99. T h e n t h e r e is an ~? > 0 s u c h t h a t f ( a + h) - f ( a ) >-- 0 w h e n e v e r [[h[[ -----~. Let 9 > 0. Again w e a p p l y T a y l o r ' s t h e o r e m , a n d find a p o s i t i v e 8 --< ~? s u c h t h a t

<--f(a + h) - f ( a )

-9

- ~ . d P f ( a ) ( h p) <-- e[[h[~~

w h e n e v e r []hll < & So f o r 0 < ]]h]] < ~ w e h a v e

0 <--f(a + h) - f ( a )

<-- l d p f ( a ) ( h P )

+ 9

IJ.

h e n c e 0 --< dPf(a)(hP) + p!e]]h[~. N o w fix h. C h o o s e a t r 0 w i t h ]]thll < 8, w h i c h i m p l ie s dPf(a)((th) p) + p!~llthll ~ -> 0. B e c a u s e p is n o w a s s u m e d to b e e v e n w e h a v e tp > O, h e n c e dPf(a)(l~) + p!9 >- O. Bec a u s e this h o l d s f o r e a c h 9 > 0 w e c o n c l u d e that

dPf(a)(hP) >-- O. (3) We d e a l w i t h t h e c a s e w h e r e dPf(a)(hP) > 0 w h e n e v e r h r 0. D e n o t e b y S t h e unit s p h e r e in ~m. S i n c e S is c o m p a c t , t h e c o n t i n u o u s m a p ~b:S--~ ~ d e f i n e d by d~(h) := dPf(a)(hP) attains its m i n i m u m . So t h e r e is a h > 0 w i t h dPf(a)(hP) >-- h w h e n e v e r Ith[I = 1, so dPf(a)(h p) >-- h[[h[~~ f o r all h ~ ~m. Let 0 < 9 < h/p!. Inv o k i n g T a y l o r ' s t h e o r e m for a last time, w e c h o o s e a 8 > 0 s u c h t h a t !f(a + h) - f ( a ) - ~dPf(a)(hP)l <9][h[~~ w h e n e v e r [Ih[[-< & F o r t h e s e tt Pwe t h e n h a v e -e~[h]~~ <--f ( a + h) - f ( a ) - ~ d ~ a ) ( h p) <-- d]hl~, h e n c e 0 <

o

+

h

IIhlP <-f(a + h) - f ( a ) .

But this m e a n s t h a t f ( a + h ) > f ( a ) Ilhll <

as l o n g as 0 <

D

The High-Brow Approach Of course, it is a e s t h e t i c a l l y u n s a t i s f a c t o r y to h a v e a condition s u c h as dPf(a)(h p) r 0 w h e r e o n e w o u l d e x p e c t j u s t dPf(a) r O. But this c a n be t a k e n care o f if w e v i e w h i g h e r derivatives as m u l t i l i n e a r maps. So let E, F be finite-dimensional v e c t o r s p a c e s ( o v e r the reals), a n d c h o o s e n o r m s in E, F. I f D C E is open, a m a p f : D --~ F is said to b e differ-

50

THE MATHEMATICALINTELLIGENCER

One p r o v e s i m m e d i a t e l y that T in the above definition is uniquely d e t e r m i n e d an d c o n s e q u e n t l y writes df(a) := T. One says that f i R c o n t i n u o u s l y differentiable on D iff i R differentiable at e a c h x ~ D and d f : D ~ L(E, F) is continuous. ff El, . . . , Ek are v e c t o r spaces, the space o f k-multilinear m a p s of E1 . . . . , Ek to F is d e n o t e d by L ( E 1 , . . . , Ek; F). We e n d o w this s p a c e with the o b v i o u s norm. If E1 = .... Ek = E this s p a c e is d e n o t e d Lk(E; F). If h E E w e write T(h k) := T(h, . . . , h). By Lk(E; F) one d e n o t e s t h e subspace of s y m m e t r i c mappings. We say that T E Lk(E; ~ ) is p o s i t i v e ( n e g a t i v e ) s e m i d e f i n i t e if T(h k) >- 0 (resp. ~ 0) for all h ~ E. Similarly, T is said to be p o s i t i v e ( n e g a t i v e ) definite if T(h k) > 0 (resp. T(h k) < 0)) w h e n e v e r h r 0. Returning to differentiation, w e k n o w f r o m (multi-)linear algebra that t h e r e ar e n a t u r a l i s o m o r p h i s m s

L(E1, L(E2, . . . , Ek; F ) ) ~ L(E1, . . . , Ek-1; L(Ek, F ) ) ~-- L(E1, . . . , Ek; F). If f is differentiable o n D and d f : D--> L(E, F) is a g a i n differentiable at a E D, t h e n d2f(a) := d ( d f ) ( a ) ~ L(E; L(E, F)) -~ L2(E; F). Inductively, w e define dkf(a) : = d(d k l f ) ( a ) E Lk(E; F ) if d k - l f : D --) L k 1 (E; F) is differentiable at a. Th e S c h w a r z l e m m a t h e n tells us that in f a c t dkf(a) ~ Lks(E; F) p r o v i d e d f i R k t i m e s c o n t i n u o u s l y differentiable. Our t h e o r e m t h e n l o o k s a l m o s t the same: THEOREM. L e t U be a n o p e n s u b s e t o f a f i n i t e - d i m e n s i o n a l vector space E, a n d let f : U--> ~ be n t i m e s c o n t i n u o u s l y differentiable. L e t 2 <--p <-- n, a n d a s s u m e that f o r s o m e a E U w e have dkf(a) = Of o r 1 <-- k <--p - 1 but dPf(a) r O. Then the f o l l o w i n g holds: (1) A n e c e s s a r y c o n d i t i o n f o r f to h a v e a local e x t r e m u m

at a i s that p be even. L e t then p be even. (2) A n e c e s s a r y c o n d i t i o n f o r f to have a mum (minimum) at a i s that d~f(a) (positive) semidefinite. (3) A s u f f i c i e n t c o n d i t i o n f o r f to have a mum (minimum) at a i s that dPf(a) (positive) definite.

local m a x i be n e g a t i v e local m a x i be n e g a t i v e

The p r o o f carries o v e r a l m o s t v e r b a t i m f r o m the lowb r o w case. Th er e is b u t o n e crucial point: We k n o w that dPf(a) r 0, so w e k n o w that t h e r e are hi, 9 9 9 , hp ~ E w i t h d P f ( a ) ( h l , . . . , hp) r 0; b u t w e n e e d to k n o w that this happens with hi, 9 hp all equal. It is h e r e that w e exploit t he fact that dPf(a) is s y m m e t r i c . If p = 2 w e could use the parallelogram identity to c o n c l u d e that d2f(a) = 0 iff d2f(a)(h 2) = 0 for all h ~ E . F o r the general c a s e w e could appeal to the p o l a r i z a t i o n identity [AMR, P r o p o s i t i o n 2.2.11]: PROPOSITION. L e t E, F be f i n i t e - d i m e n s i o n a l vector s p a c e s a n d k E ~. F o r A E Lk(E; F) d e f i n e .2i : E--> F by .2i(h) : =

A ( h k) a n d denote by Sk(E; F) the vector space {fi A E L k (E; F ) } e n d o w e d w i t h the n o r m I ill : = s u p { l ~ ( h b l l IIhll <1}. T h e n ]tAll-< k*!l ~ I f o r A E Lk(E; F), a n d ^ w h e ~ res b ' i e t e d to Lk(E; F ) i s a n i s o m o r p h i s m . If y o u w a n t to p r o v e t h e t h e o r e m o n e x t r e m a in a calculus c o u r s e y o u m i g h t b e r e l u c t a n t to b o t h e r y o u r s t u d e n t s w i t h t o o m u c h m u l t i l m e a r algebra, so h e r e is a s h u p l e and direct p r o o f w h i c h I l e a r n e d f r o m m y c o l l e a g u e T h o m a s M e i x n e r [M]: LEMMA. L e t E, F be v e c t o r spaces, n E ~ , a n d T E L~'(E; F). I f T ( h ~*) = 0 f o r all h ~ E, t h e n T = O. PROOF. W e p r o c e e d b y i n d u c t i o n . If n = 1 t h e r e is n o t h i n g to p r o v e . So a s s u m e t h e c l a i m for n - 1 a n d let T E L~(E; F) C L(E, L~ I(E; F)). S u p p o s e T(h '~) = 0 for all h b u t T r 0. T h e n t h e r e m u s t b e a n a E E w i t h 0 r T(a) E L~*-I(E; F). S i n c e we assume the claim for n - 1 there must be a b E E with T(a, b, . . . , b) r O. By o u r a s s u m p t i o n , w e m u s t h a v e t h a t T((Aa + b) ~2) = 0. E x p a n d i n g b y m u l t i l i n e a r i t y a n d using t h e s y m m e t r y o f T to c o l l e c t c o m m o n t e r m s , w e find that

'~)

0=T((Xa+b)

for allZE

= ~. mjT(a,...,a, j

0

= ~

b,...,b)A

j times

j

withmi~N\{0}

REFERENCES

n - - j times

m j T ( a , . . . , a, b . . . , b)M,

j--1

T(b'9 = T(a'9 = O.

because

Now we put A = 1,..., equation:

(*)

...

22

i

n

n - 1 and we obtain the following

,...

2

(n -

1) 2

1)

[AMR] R. Abraham, J.E. Marsden, T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, Springer, 1988, AppL Math. Sci. 25. [J] J0rgen Jost, Postmodern Analysis, Universitext, Springer, Berlin et al., 1998. [M] Thomas Meixner, Personal communication.

2''-1

"..

:

...

( n - ' 1 ) '* 1

i

1

Of course, we now might wish to remove the assumption that our vector spaces be finite-dimensional. So now a s s u m e t h a t E is a B a n a c h s p a c e . T h e n w e m a y c o p y t h e above arguments with almost no changes: We simply c h a n g e o u r n o t a t i o n : w e n o w d e n o t e b y L ( E , F) a n d Lk(E; F ) t h e s p a c e o f all c o n t i n u o u s ( k - ) l i n e a r m a p s (in t h e finite-dimensional case (multi-)linear maps are automatic a l l y c o n t i n u o u s , s o t h i s is e v e n c o n s i s t e n t w i t h o u r p r e v i o u s t e r m i n o l o g y ) . M o r e o v e r , I h a v e d e l i b e r a t e l y c h o s e n refe r e n c e s ([J] a n d [AMR]) t h a t a c t u a l l y d e a l w i t h t h e infinite-dimensional case, and Meixner's lemma does work in a n y v e c t o r s p a c e . T h e r e is b u t o n e p o i n t w h e r e w e really n e e d e d a f i n i t e - d i m e n s i o n a l v e c t o r s p a c e : in p a r t 3) o f t h e p r o o f w e u s e d t h e f a c t t h a t t h e u n i t s p h e r e is c o m p a c t . T h e b e s t w a y o u t o f t h i s is to r e q u i r e j u s t w h a t w e n e e d : Let u s c a l l T C Lk(E; R) s t r o n g l y p o s i t i v e ( n e g a t i v e ) deftn i t e if t h e r e is a A > 0 s u c h t h a t T(h k) > )tllhllk (resp., T(h k) <- -Alibi] k) f o r all h E E. T h e h i g h - b r o w t h e o r e m t h e n c o n t i n u e s t o h o l d if w e r e p l a c e " f i n i t e - d i m e n s i o n a l v e c t o r space" by "Banach space," provided we insert "strongly" b e f o r e " n e g a t i v e ( p o s i t i v e ) d e f i n i t e " in 3).

m 2 T ( a , a, b, •

, b, b) ~

=

m,~ iT(a, a, a,

.

. , a, b ) /

We d e n o t e t h e m a t r i x o n t h e l e f t - h a n d s i d e b y M , _ 1 a n d c l a i m t h a t d e t M R - 1 r O: S t a r t i n g f r o m t h e l a s t c o l u m n , m u l tiply t h e (3" - 1)-st c o l u m n b y n - 1 a n d s u b t r a c t t h e r e s u l t f r o m t h e j - t h c o l u m n . T h u s t h e first c o l u m n r e m a i n s unaltered. This gives us

det

2 ~

22(3 - n ) ~

... '..

2

( n - 2 ) ( - 1)

...

1

0

..

2 ~ 2(3 - n ) i (n - 2)'-2(-

1)

0

Expanding the determinant according to the last row and collecting common factors we obtain ( - 1 ) ' ~ ( n - 1 ) ( - 1 ) '~ 2(n - 2)! d e t M,~-2 = ( n - 1)! detM,~ 2. B e c a u s e d e t M2 r 0, t h i s s h o w s w h a t w e n e e d e d . B u t t h e n (*) h a s o n l y t h e t r i v i a l s o l u t i o n . In p a r t i c u l a r , m l T ( a , b, . . . . b) = 0, w h i c h s h o w s t h a t T(a, b, . . . , b) = O. C o n t r a diction. []

VOLUME 25, NUMBER 1,2003

51