") The vector has norm one while gradient and hessian of are@ 0
f0 œ Ð/ ß / Ñ 0 œ / /
/ /
BC BC BC BC
BC BC
; . [ Thus
W@@0 ÐSÑ œ @ †X [0 ÐSÑ † @ œ
Ð ß Ñ † " " † œ # œ
" "
cos sin cos cos cos sin sin
α α sinα α α α α
α # #
" sin#α. The condition is satisfied iff sin# œ !α and this is true for αœ ! ”αœ Î# ”1 αœ1”αœ $ Î#.1
# 0) is differentiable at point Ð!ß !Ñ iff exist an affine function X ÐBß CÑ such that:
ÐBß CÑ637Ð ß Ñ
0 ÐBß CÑ 0 Ð!ß !Ñ X ÐBß CÑ
B C œ !
Ä ! ! # # . Rewritten the limit in polar coordinates with X ÐBß CÑ œ +B ,C, we have
3637
3 α 3 α
3
Ä !
3 α3 α 3 α3 α
3
cos sin l cos sin l
# Ð+ , Ñ
cos sin œ
3637 3 α α α α α α
Ä ! cos sin lcos sin l Ð+cos ,sin Ñ and the limit is only if! + œ , œ !. To conclude that is differentiable at point 0 Ð!ß !Ñ we must verify that the result is uniformly on , for this goal considerα
l3cos sinα αlcos sinα αll œ Ð3 cos sinα αÑ Ÿ ß a# 3 α and if + œ , œ ! we can conclude that a !ß% 3 Ê l% 3cos sinα αlcos sinα αll %, convergence is uniformly and is differentiable at point 0 Ð!ß !Ñ.
$) Lagrange function of the problem is
_ÐBß Cß Dß Ñ œ- B" C" "D ÐB C D $Ñ- # # # .
f œ _ B"# # Bß - C"# # Cß - D"# # Dß ÐB C D $Ñ- # # # . Put the first order condition we have the sistem:
# B œ !
# C œ !
# D œ ! B C D œ $
Ê Ê
" # B œ !
" # C œ !
" # D œ ! B C D œ $
B œ "Î#
C œ "Î#
D œ "Î#
"
B"
C
"
# D # #
$
$
$
# # #
#
#
#
$
$
$
- - -
- - -
- - - B C D œ $
Ê
# # #
B œ "Î#
C œ "Î#
D œ "Î#
$ "Î% œ $ Ê
B œ …"
C œ …"
D œ …"
œ „"Î#
$
$
$
$
- - -
-# -
. Two points satisfied first order conditions
T œ Ð"ß "ß "ß "Î#Ñ" and T œ Ð "ß "ß "ß "Î#Ñ# .
[ -
-
- µ œ
! #B #C #D
#B # ! !
#C ! # !
#D ! ! #
#
B #
C
# D
$
$
$
, becouse there's only one constraint
we must analyse third and fourth principal minor of border hessian:
[ -
- -
-
µ œ œ #B #C œ
! #B #C
#B # !
#C ! #
#B !
#C #
#B #
#C !
$ #
B
# C
# C
# B
$
$
$
$
#B#C#$ #- #C#B#$ #- ;
[ -
-
-
- µ -
œ œ #D
! #B #C #D
#B # ! !
#C ! # !
#D ! ! #
#B # !
#C ! #
#D ! !
%
#
B #
C
# D
# B
# C
$
$
$
$
$
# µ œ #D # ! # µ œ
! #
D# $ # D# $
#
B #
C
$ $
$
$
- [ - - [
-
#D#B#$ #-C#$ #- D#$ #- #B#C#$ #- #C#B#$ #-œ
#B#C#$ #-D#$ #- #C#B#$ #-D#$ #- #D#B#$ #-C#$ #- . Second order conditions:
[µ [µ
ÐT Ñ œ "# ! ÐT Ñ œ &% ! T
$ " ; . % " " minimum.
[µ [µ
ÐT Ñ œ "# ! ÐT Ñ œ &% ! T
$ # ; . % # # maximum.
The proposed problem has minimum equal on point $ Ð"ß "ß "Ñ and maximum equal
$ on point Ð "ß "ß "Ñ.
%) The condition is satisfied on point , T " † / œ " † /$ $; rewritten the condition as B /#C D# # C /B #D# # œ ! we calculate the gradient of
0 ÐBß Cß DÑ œ B /#C D# # C /B #D# #.
f0 œ Ð/#C D# # #BC/B #D# #ß %BC/#C D# # /B #D# #ß #BD/#C D# # %CD /B #D# #Ñ and f0 ÐT Ñ œ Ð / ß $ / ß # / Ñ$ $ $ , but any element of f0 ÐT Ñ is different from zero thus we can have in implicy form a function of kind ÐBß CÑ È DÐBß CÑ or
ÐBß DÑ È CÐBß DÑ or ÐCß DÑ È BÐCß DÑ. If we choose the first kind,
fDÐ"ß "Ñ œ 0 ÐT Ñ0 ÐT ÑDBww ß 0 ÐT Ñ0 ÐT ÑCDww œ # //$$ß # /$ /$$ œ ß# #" $. The equation of tangent plane will be
D " œ fDÐ"ß "Ñ † B " Í D " œ ÐB "Ñ ÐC "Ñ Í C "
"# $#
B $ C # D œ !.
& fD œ Ð$B Cß B $C Ñ D œ 'B D Ð"ß !Ñ œ D Ð!ß "Ñ œ '
) , # α α # [ 'Cα . BBww CCww ,
α
D Ð"ß !Ñ œ D Ð!ß "Ñ œwwBC CBww α; the condition is true when $' œα# Êαœ „'.
