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 cal...
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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
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( 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