ԵՐԵՎԱՆԻ ՊԵՏԱԿԱՆ ՀԱՄԱԼՍԱՐԱՆ

Կ. Լ. ԱՎԵՏԻՍՅԱՆ

ԼՈԿԱԼ ԵՎ ՊԱՅՄԱՆԱԿԱՆ

ԷՔՍՏՐԵՄՈՒՄՆԵՐ

ԵՐԵՎԱՆ

ԵՊՀ ՀՐԱՏԱՐԱԿՉՈՒԹՅՈՒՆ

517(075.8) 22.161 73 Հրատարակության է երաշխավորել ԵՊՀ մաթեմատիկայի և մեխանիկայի ֆակուլտետի գիտական խորհուրդը

Ավետիսյան Կ. Լ. 791 Լոկալ և պայմանական էքստրեմում եր/Կ. Լ. Ավետիսյան. -Եր.: ԵՊՀ հրատ., 2018, 132 էջ։ Ձեռնարկում շարադրված է մաթեմատիկական անալիզի կիրառական ուղղություններից մեկը՝ մի քանի փոփոխականի ֆունկցիաների էքստրեմումների տեսությունը։ Ներկայացված են լոկալ (բացարձակ) և պայմանական էքստրեմումների որոնման հիմնական մեթոդներն ու առանձնահատկությունները։ Բացի անհրաժեշտ տեսական նյութից ձեռնարկը պարունակում է 35 մանրամասն վերլուծված օրինակներ, ինչպես նաև ինքնուրույն աշխատանքի համար 60 վարժություններ իրենց պատասխաններով։ Ձեռնարկը նախատեսված է մաթեմատիկական, տնտեսագիտական և բնագիտական մասնագիտությունների ուսանողների համար։

517(075.8) 22.161 73

ISBN 978-5-8084-2279-7

 ԵՊՀ հրատ., 2018  Ավետիսյան Կ. Լ., 2018

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a11 a12 . . . a1n

⎟⎜ ⎜ a21 a22 . . . a2n ⎟ ⎟⎜ ⎜ ⎟ .. .. . . . ⎜ . .. ⎟ . . ⎠⎝ an1 an2 . . . ann ⎛

⎞ y1 y2 .. .

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

⎞ y1

⎜ ⎟ ⎜ y2 ⎟ ⎜ ⎟ & y = ⎜ . ⎟ ⎜ .. ⎟ ⎝ ⎠ yn

;, +, 26 1

Q ≡ Q(y) ≡ Q(y1 , y2 , . . . , yn ) =

aij yi yj = y T A y

i,j=1

<# 1? Q(y) ?

$ % Q(y)5

0) " y = 0

' 0 y1 , y2 , . . . , yn %%. ( " 0

Q(y) > 0,

∀y=0

2? Q(y) ? <

y12 + · · · + yn2 = 0 :

% Q(y)5

0) "

y = 0 ' 0 y1 , y2 , . . . , yn %%.(

" Q(y) < 0,

3? Q(y) ? $

∀y=0

& % ,

%- Q(y)5

0) y ∈ R

y12 + · · · + yn2 = 0 :

" 5 " y ∈ R n

' Q(y) ≥ 0,

∀ y ∈ Rn ,

4? Q(y) ? <

∃ y ∈ Rn , y = 0

s.t.

Q(y ) = 0 :

& $ % ,

%- Q(y)5

0) y ∈ R ' n

" 5 " y ∈ Rn ' Q(y) ≤ 0,

∀ y ∈ Rn ,

5? Q(y) ? <

∃ y ∈ Rn , y = 0

s.t.

Q(y ) = 0 :

%% (

0 ) # 6? Q(y) ? < %

' %

7? Q(y) ? <

%( , %%-

! ! ) 0

# def

12 Q(y1 , y2 , y3 ) = y12 + 2y22 + 3y32

C& * 15

) Q(y1 , y2 , y3 ) > 0

< (

∀ y = (y1 , y2 , y3 ), y12 + y22 + y32 = 0#

def

22 Q(y1 , y2 , y3 ) = y12 + y22 + y32 + 2y1 y2 + 2y1 y3 + 2y2 y3

< ) Q(y1 , y2 , y3 ) = (y1 + y2 + y3 )2 ≥ 0

∀ y = (y1 , y2 , y3 ) ∈ R3

Q(y1 , y2 , y3 ) = 0 0 y1 = −y2 − y3 = 0# def

32 Q(y1 , y2 , y3 ) = y12 − y22 + y32 + y1 y2 + y1 y3 + y2 y3

%. Q(1, 0, 0) = 1 > 0, Q(0, 1, 0) = −1 < 0# W - = % <

R $ - n

def

Q ≡ Q(y) ≡ Q(y1 , y2 , . . . , yn ) =

aij yi yj ,

aij = aji ,

i,j=1

$ ⎞

⎛ ⎜ ⎜ ⎜ A = (aij ) = ⎜ ⎜ ⎝

a11 a12 . . . a1n

⎟ a21 a22 . . . a2n ⎟ ⎟ : .. .. . . .. ⎟ . . . . ⎟ ⎠ an1 an2 . . . ann

+2@9.

< %& def

Δ1 = a11 ,

def

Δ2 =

a11 a12 a21 a22

a11 a12 a13 def

a21 a22 a23 ,

Δ3 =

a31 a32 a33

a11 a12 . . . a1k def

Δk =

a11 a12 . . . a1n

a21 a22 . . . a2k .. .. . . . , . .. . .

a21 a22 . . . a2n .. .. . . . , . .. . .

def

Δn =

an1 an2 . . . ann

ak1 ak2 . . . akk

A , &7 * +

+ $4 %&? ,%&& %&, 27 ,A) ' 1? M! " +2@9. ? < ) ) ' 0 0)

) Δ1 > 0,

Δ2 > 0,

Δ3 > 0,

, Δn > 0 :

+2@10.

2? M! " +2@9. ? < 0 ) ) '(

'%. )

) 0(

Δ1 < 0,

Δ2 > 0,

Δ3 < 0,

, (−1)n Δn > 0 :

+2@11.

3? M! " +2@9. ? < 0 ) , )- ) ' 0)

Δ1 ≥ 0,

0 ) Δ2 ≥ 0,

Δ3 ≥ 0,

0 5 k5 ' #

, Δn ≥ 0,

+2@12.

∃ Δk =

4? M! " +2@9. ? < ) ,0 )- ) '

0 '(

) ! Δ1 ≤ 0,

Δ2 ≥ 0,

Δ3 ≤ 0,

, (−1)n Δn ≥ 0,

+2@13.

∃ Δk =

0 5 k5 ' # 5? C A

+2@11. +2@12. +2@13. !

+2@10.

0 ) !

! +2@9. ? < %. , - # A%(+. + 26 L

? A = (aij )

) A'

) )

# U"! aij ) Q ≡ Q(y) ≡ Q(y1 , y2 , . . . , yn ) =

(

" )

aij yi yj = y T A y

i,j=1

< ) H 2275 1)5 !

%* 4 9: &,+, ! & & -

, / , : 5%5% * 6+*7 & ; , & >) &+ +?

P9 < 2@6> $

" #

- V 2@6>%

%&, 28 1 M0 (x01 , x02 , . . . , x0n ) 5 U (M0 ) f (M )

25 f (M ) ∈ C U (M0 ) M0

df (M0 ) ≡ 0# &

f (M ) M0

f (M )

def

aij =

∂ 2 f (M0 ) , ∂xi ∂xj

)

i, j = 1, n,

Q ≡ Q(dx1 , dx2 , . . . , dxn ) ≡ d f (M0 ) =

aij dxi dxj

+2@14.

i,j=1

< A = (aij )

1? C Q ? < ) ! M0 f (M ) ,. - ))

M0 = Mmin #

2? C Q ? < 0 ) ! M0 M0 = Mmax #

,. - ))

f (M )

3? C Q ? < %. , - ! M0 f (M )

))

4? C Q ? < ) ! ) 2

) ) ! ) ) ) ,

! f 5

!' )

V 2@7>

$ < %&

, #

% %&, 29 1 M0 (x01 , x02 , . . . , x0n ) 5 U (M0 ) f (M )

25 f (M ) ∈ C 2 U (M0 ) M0

f (M )

df (M0 ) ≡ 0# &

25 )

f (M )

∂ 2 f (M0 ) , i, j = 1, n, ∂xi ∂xj " A = (aij ) # def

aij =

1? C A 0) Δ1 > 0,

Δ2 > 0,

! M0 f (M )

Δ3 > 0,

, Δn > 0,

,. - ))

+2@15.

M0 = Mmin # 2? C A

Δ1 < 0,

'%.

Δ2 > 0,

! M0 f (M )

Δ3 < 0,

, (−1)n Δn > 0,

,. - ))

+2@16.

M0 = Mmax # 3? C Q ? < %. , - ! M0 f (M )

))

4? 1

! Δ1 ≥ 0,

Δ2 ≥ 0,

Δ3 ≥ 0,

, Δn ≥ 0,

+2@17.

, (−1)n Δn ≥ 0,

+2@18.

Δ1 ≤ 0,

Δ2 ≥ 0,

Δ3 ≤ 0,

0 5 k5 ' # : ! M0 ) ) ,

∃ Δk =

) 2

! f 5

) ) ! )

!' ) '"

5? C A

+2@16. +2@17. +2@18. ! ! M0 f (M )

+2@15.

0 ) ! ))

* V 2@9>% n>'

''( ," V 1@7>%

$ %

C& * 16 E def

u = f (M ) = f (x, y, z) = 2x2 − xy + 2xz − y + y 3 + z 2 : R3 −→ R

# +.

∞

) 0

f ∈ C (R ) (

' . ? ) # E

' " ) ') ' ⎧ f = 4x − y + 2z = 0 ⎪ ⎪ ⎨ x ⎪ ⎪ ⎩

fy = −x − 1 + 3y 2 = 0 fz = 2x + 2z = 0 :

D ) ' "

' 1 1 1 1 2 1 M1 , ,− , M2 − , − , : 3 3 3 4 2 4

A f # :

# 7 f

)

0 !

25 )

fxx (M ) = 4,

fxy (M ) = −1,

fyy (M ) = 6y,

fxz (M ) = 2,

fzz (M ) = 2,

fyz (M ) = 0 :

$" f

25 )

< d2 f (M ) = fxx (M ) dx2 + fyy (M ) dy 2 + fzz (M ) dz 2 + + 2fxy (M ) dx dy + 2fxz (M ) dx dz + 2fyz (M ) dy dz =

= 4 dx2 + 6y dy 2 + 2 dz 2 − 2 dx dy + 4 dx dz,

⎛

−1 2

⎞

⎜ ') A ≡ A(M ) = ⎜ ⎝ −1 6y 7"

. M1

0 !

⎟ 0 ⎟ ⎠: 0 2

1 2 , , − 13 3 3

# ;?

') ⎛

⎜ A(M1 ) = ⎜ ⎝ −1

−1 2

7 A(M1 ) Δ1 = 4 > 0,

Δ2 =

−1

d2 f (M ) <

⎞

⎟ 0 ⎟ ⎠:

,).- −1 2

Δ3 =

= 15 > 0,

−1

= 14 > 0 : +2@19.

; A(M1 )

! ' <

2295

M1 f

))

Mmin = M1 #

+ +2@19. !

A(M1 )

0) ) ) A) ' ,H 227-

A(M1 )

' ! .

d f (M1 ) < ) ' < H( 2285

! '

M1 f

Mmin = M1 #

))

: '" M2 − 14 , − 12 , 14 0 !

d f (M ) <

# ;?

') ⎛

−1 2

⎞

⎜ ⎟ ⎟ A(M2 ) = ⎜ ⎝ −1 −3 0 ⎠ : 0 2 7 A(M2 )

Δ1 = 4 > 0,

Δ2 =

−1

−1 −3

,).- −1 2

= −13 < 0,

= −14 < 0 :

−1 −3 0

Δ3 =

+2@20.

+2@20. !

A(M2 )

) = ) ) 0 !

) 0

2295

(

2295

1)4

55 !

' ) d2 f (M2 )

+ " ) <

d2 f (M2 ) = fxx (M2 ) dx2 + fyy (M2 ) dy 2 + fzz (M2 ) dz 2 + + 2fxy (M2 ) dx dy + 2fxz (M2 ) dx dz + 2fyz (M2 ) dy dz =

= 4 dx2 − 3 dy 2 + 2 dz 2 − 2 dx dy + 4 dx dz : L d2 f (M2 ) ? < %. #

dy = dz = 0 dx = 0 ! d2 f (M2 ) = 4 dx2 > 0# dx = dz = 0 dy = 0 ! d2 f (M2 ) = −3 dy 2 < 0# ;? d2 f (M2 ) < %. H 2285 <)

34%

N+*7

>; & * ? 9: &,+,&/=

$& < ,.%L/ 9(

,$- 9(

S , $&

+ , .

OL& L& ':/ &*+ 5%5% * 6+*7 & ; , &

D!,

$ % ! 6 #

C& * 17 1 def

z = f (x, y) =

1 − x2 − y 2

D := {(x, y) : x2 + y 2 ≤ 1} %

# :

'") K 15

))

!") f

"0 )) (

Mmax = (0, 0) # A )) 0<

. ) ) # :

% ) . <! # E ) ) f

0 D

) D5

5 0"

5 ' B

y = 1 − x

MT

' def

ϕ(x, y) = x + y − 1 = 0 ,

* # N

* )

+3@1. .

) !

⎧ ⎨ f (x, y) = 1 − x2 − y 2 −→ extr. ⎩ϕ(x, y) ≡ x + y − 1 = 0

) ) !

+3@1. ! ' ? ! ) )

! R '

! ' !

+3@2. +3@2. . (

+3@2. .

ϕ(x, y) = 0

+3@1.

+ ϕ(x, y) = 0 0< ! ! ! 0<

Mmax = (0, 0) '# A

+3@2. . +3@1. ! '

) #

; % )

ϕ(x, y) ≡ x+y−1 = 0

+3@2. . ' 0 ) ! y = 1−x '

= f (x, 1 − x) =

f (x, y)

0 f

(

0 < <(

1 − x2 − (1 − x)2 −→ extr.,

x ∈ [0, 1] :

y=1−x

A %

. )

) f (x, 1 − x) =

1 − x2 − (1 − x)2 = 2x(1 − x),

: ) Mmax = √

( 12 , 12 )

xmax =

+3@1. ! ' 0 !

) fmax =

f ( 12 , 12 )

( % , #

" , ) $ , -

< "

, y>% ,$ $ x> $& y = y(x) ϕ(x, y) = 0 %

% ! ϕ(x, y) = 0

$ ⎧ ⎨x = x(t) ⇐⇒ ⎩ y = y(t)

t 0 ≤ t ≤ t1 ,

" " % <$" z = f x(t), y(t)

−→ extr.

t0 ≤ t ≤ t1 ,

+3@3.

) $ (

t ( ''( $ -

+3@2. ( % 9

+3@3.

( = ,

, ϕ(x, y) = 0 $ " x y ''(

$ % ,$

$ R-

ϕ(x, y) = 0 %

''(#

y = y(x) ) $ $ $ 0' ϕ(x, y) = 0

⇐⇒

y = y(x)

,$ ) $ :

V y = y(x) ) $ ,$ 4 - $% ,

z = f (x, y) ) $

''( ) $

! %& z = f x, y(x)

−→ extr.

x 0 ≤ x ≤ x1 :

A! %

+3@4.

+3@4. , ) $

$ +, ) $ $ .& zx = fx + fy yx :

+3@5.

= "$ y = y(x) ,$ ) $ $ % % ,$ ) $ $ , -

$ ϕ(x, y) = 0 %

% x>& y>% ( x>$& d ϕ x, y(x) = ϕx + ϕy yx : +3@6. 0= dx $$ ϕy = 0 ,$ ) $ $ %& yx = −

ϕx : ϕy

+3@7.

+3@7.>% +3@5.> +3@4. $ %& zx = fx − fy Y %

$ %

z , ) $

ϕx : ϕy

+3@8.

(

& +3@8. $ % $ ! &

zx = fx − fy

ϕx =0: ϕy

+3@9.

= % , +3@9.

fx − fy <$

ϕx = 0, ϕy

+3@10.

ϕ(x, y) = 0 :

9 %

' R 64 def

−λ =

fy fx = , ϕx ϕy

" (−) % P9 λ &

+3@11.

! (

$ +3@10. 9% 9 ⎧ f (x, y) + λϕx (x, y) = 0 ⎪ ⎪ ⎨ x ⎪ ⎪ ⎩

fy (x, y) + λϕy (x, y) = 0

+3@12.

ϕ(x, y) = 0 :

= +3@12. 9%

f (x, y) ) $

;& <) -

, / ϕ(x, y) = 0 # A '- ⎧ ⎨f (x, y) −→ extr. ⎩ϕ(x, y) = 0

+3@13.

( % ;! +3@12. 9%&

, % +# $ %. $$

% +3@12. 9 '' ! # 64 ) $ & def

L(x, y) ≡ L(x, y, λ) = f (x, y) + λϕ(x, y) ,

< L ) $ $& < ! P, - *Q

λ ∈ R, +3@14.

+ . D ⊂ R2

" (x, y) % ''( λ>

(x, y) ∈ D,

$& < 6+*7 λ #

$& < %&%) $%&' *7

9 " %

$& < ,.%L A +3@12. 9 ,

% 9 4 L ) $ $ & ⎧ ⎧ fx (x, y) + λϕx (x, y) = 0 L (x, y, λ) = 0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ x ⇐⇒ fy (x, y) + λϕy (x, y) = 0 Ly (x, y, λ) = 0 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ϕ(x, y) = 0 Lλ (x, y, λ) = 0 : ? ' $$$

+3@15.

% % 9 4 ) $ $ - % # %&, 31 1 z = f (x, y) ϕ(x, y) Ω ⊂ R2 f 5 !

f (x, y), ϕ(x, y) ∈ C 1 (Ω) M0 (x0 , y0 ) ∈ Ω

ϕ(x, y) = 0 ! '(

& M0 (x0 , y0 )

" ) " #

ϕ(x, y)

ϕx (x0 , y0 )

ϕy (x0 , y0 )

(

: !

? 0

λ0 ∈ R ! (x0 , y0 , λ0 ) +3@15. ' D

# / ,

+3@16. ( %

! 4 4 ,

% 9 4 9 ,

'' - 9 4 ) $

f (x, y) 6+*7

* 9:

L&/ ; , &<: 9

f R $& < 6+*7

* 4%* 4 9: = 4 N 9 -

L&=

% %&, 32 ,U 1

25 f (x, y)

0 ! !

. ϕ(x, y) = 0 ! ' ! ⎧ ⎨f (x, y) −→ extr. ⎩ϕ(x, y) = 0 :

1 M0 (x0 , y0 ) 5 λ = λ0 ! D

⎪ ⎪ ⎩ $" D

fy (x0 , y0 ) + λ0 ϕy (x0 , y0 ) = 0

+3@16. (

f (x, y)

L(x, y) ≡ L(x, y, λ) = f (x, y) + λϕ(x, y)

dL(x0 , y0 ) ≡ 0 ⎧ f (x , y ) + λ0 ϕx (x0 , y0 ) = 0 ⎪ ⎪ ⎨ x 0 0

(

+3@17.

ϕ(x0 , y0 ) = 0 : 25 ) M0 (x0 , y0 )

d2 L(x0 , y0 ) = Lxx (x0 , y0 ) dx2 + 2Lxy (x0 , y0 ) dx dy + Lyy (x0 , y0 ) dy 2 :

1? C

d2 L(x0 , y0 ) > 0

∀ dx, dy

) ,*-

" 0

dx2 + dy 2 = 0,

ϕx (x0 , y0 ) dx + ϕy (x0 , y0 ) dy = 0 ,

! M0 (x0 , y0 ) f

+3@18.

,. - !

(

M0 (x0 , y0 ) = Mmin ϕ(x, y) = 0 ! ' 2? C

d2 L(x0 , y0 ) < 0

∀ dx, dy

) ,*-

" 0

dx2 + dy 2 = 0, ! M0 (x0 , y0 ) f

ϕx (x0 , y0 ) dx + ϕy (x0 , y0 ) dy = 0, ,. - !

(

M0 (x0 , y0 ) = Mmax ϕ(x, y) = 0 ! '

3? C d2 L(x0 , y0 ) ) %. , ( 0 - dx, dy )( ,*-

+3@18. ' %

M0 (x0 , y0 ) f

# 4? C d2 L(x0 , y0 ) ≥ 0

d2 L(x0 , y0 ) ≤ 0 0 dx, dy

) ,*- 0

+3@18. ' %(

! ) 2 M0 (x0 , y0 )

! f 5 !

! ) ) ) ,

(

) ) (

!' ) '(

= & 4

+B V

1@6.

9 ) $%

+9 4 ) $ . $

% W

! 6 $ %

,! 9 4

C& * 18 6 K 2 75

)

def

z = f (x, y) = x2 + y 2 : R2 −→ R+ ,

, )!

!0)- !")

" 0 f 5 0<

: <! 0) ) ! ϕ(x, y) = x + y − 1 = 0 ! ' ! ⎧ ⎨f (x, y) = x2 + y 2 −→ extr. ⎩ϕ(x, y) def = x+y−1=0:

Mmin = (0, 0)

+3@19.

. * ' )

y = 1 − x ! '

%%. f (x, 1 − x)

H

(

D +3@19. . D !

(

! "

$" f 5 D

L(x, y) = f (x, y) + λϕ(x, y) = x2 + y 2 + λ(x + y − 1), λ5

λ ∈ R,

D 0" ! , - # +.

D L(x, y)

# 1 '(

" ) ') ' ⎧ L (x, y, λ) = 2x + λ = 0 ⎪ ⎪ ⎨ x ⎪ ⎪ ⎩

Ly (x, y, λ) = 2y + λ = 0

Lλ (x, y, λ) = x + y − 1 = 0 :

7 15 25 '

) x = y

⎧ ⎨λ = −1

⎩x = y = 1 : : ! M0 ( 12 , 12 ) D L(x, y)

+3@20.

+3@19. . ! '

:

! # 7 D L(x, y) )

M0 ( 12 , 12 )

Lxx (x, y)

1 1 , = 2, 2 2 1 1 , = 0, Lxy 2 2

Lxx

= 2,

Lxy (x, y) = 0,

Lyy (x, y)

1 1 , 2 2

Lxx

Lyy

= 2,

1 1 , 2 2

dx +

2Lxy

= 2 dx2 + 2 dy 2 > 0,

D .

1 1 , 2 2

= 2,

dx dy +

Lyy

1 1 , 2 2

dy 2 =

dx2 + dy 2 = 0 :

+3@21.

25 )

0 dx2 + dy 2 = 0 H 3225 '

M0 ( 12 , 12 ) M0 ( 12 , 12 ) !

f 5 ,. - !

Mmin =

ϕ(x, y) = 0 ' !

+ H

3225 !'

+3@18.

' ? ) !

ϕx (x0 , y0 ) dx + ϕy (x0 , y0 ) dy = 0,

1 1 d2 L , = 2 dx2 + 2 dy 2 = 4 dx2 > 0 2 2

dx + dy = 0,

dx = 0,

" 0 M0 ( 12 , 12 ) f 5 ,. - !

Mmin =

M0 ( 12 , 12 )

(

ϕ(x, y) = x + y − 1 = 0

C& * 19 1 def

z = f (x, y) =

1 − x2 − y 2

D := {(x, y) : x2 + y 2 ≤ 1} %

# :

'") K 15 ?

K 175

x+y−1 = 0 ! ' !

)) ?

V !

ϕ(x, y) =

%%. !") Mmax = ( 12 , 12 ) ! :

! . ⎧ def ⎨f (x, y) = 1 − x2 − y 2 −→ extr.

+3@22.

⎩ϕ(x, y) def = x+y−1=0

) D ! " !

$" f 5 D

L(x, y) = f (x, y) + λϕ(x, y) = 1 − x2 − y 2 + λ(x + y − 1),

λ5

λ ∈ R,

D 0" ! , - # +.

D L(x, y)

# 1 '(

" ) ') ' ⎧ −x ⎪ ⎪ +λ=0 Lx (x, y, λ) = ⎪ ⎪ ⎪ 1 − x2 − y 2 ⎪ ⎨ −y Ly (x, y, λ) = +λ=0 ⎪ ⎪ 1 − x2 − y 2 ⎪ ⎪ ⎪ ⎪ ⎩L (x, y, λ) = x + y − 1 = 0 : λ 7 15 25 ' ⎧ −x ⎪ ⎨√ +λ=0 1 − x2 − x2 ⎪ ⎩x + x − 1 = 0 :

+3@22. . ! '

:

! # 7 D L(x, y) )

) x = y √ ⎧ ⎪ ⎨λ = 2 ⎪ ⎩x = y = 1 :

⎧ x ⎪ ⎨λ = √ 1 − 2x2 ⎪ ⎩x = :

: ! M0 ( 12 , 12 ) D L(x, y)

+3@23.

M0 ( 12 , 12 )

Lxx (x, y)

y2 − 1 = , (1 − x2 − y 2 )3/2

Lxx

1 1 , 2 2

√ 3 2 =− ,

Lxy (x, y)

Lyy (x, y) = d2 L

1 1 , 2 2

√

, √ 3 2 1 1 , =− , Lyy 2 2

Lxy

x2 − 1 , (1 − x2 − y 2 )3/2

=−

1 1 1 1 1 1 = Lxx , dx2 + 2Lxy , dx dy + Lyy , dy 2 = 2 2 2 2 2 2 √ √ √ 3 2 2 3 2 2 dx − 2 dx dy − dy = =− √2 =− 3 dx2 + 2 dx dy + 3 dy 2 : +3@24.

7) H 3225 !

−xy = , (1 − x2 − y 2 )3/2

!' M0 ( 12 , 12 )

dϕ(M0 ) = 0

= +3@18.

dϕ(M0 ) = ϕx (M0 ) dx + ϕy (M0 ) dy = dx + dy = 0

⇐⇒

dx = −dy :

W * ! 25 +3@24. ) √ 1 1 dL , =− 3 dx2 + 2 dx dy + 3 dy 2 = 2 2 √ √ 3 dx2 − 2 dx2 + 3 dx2 = −2 2 dx2 < 0, 0 dx = 0 : =− ; D

1 1 , 2 2

25 M0 ( 12 , 12 ) f 5 Mmax = M0 ( 12 , 12 ) ! !'

25 ) . 0( < 0 0

dx = 0 ' <

- ! ϕ(x, y) = 0

322

! )#

A%(+. + 31 C M0 (x0 , y0 ) 0 M0

)) 5

' Γ ! !" f

0 ! '(

7

'< !

.) ' % ( # & ' ! (

. B # C& * 20 :! "0 def

z = f (x, y) = (y − x2 )(y − 3x2 )

y = kx # 7 ⎧ ⎧ ⎧ ∂f ⎪ ⎪ = −2x(y − 3x2 ) − 6x(y − x2 ) = 0 ⎨x(2y − 3x2 ) = 0 ⎨x = 0 ⎨ ∂x ∂f ⎩y = 2x2 ⎩y = 0 : ⎪ ⎪ = (y − 3x2 ) + (y − x2 ) = 0 ⎩ ∂y MT

(0, 0) "0 f (x, y)

) ! f 5 0 #

f (0, 0) = 0 (0, 0) y = 0 !

f (0, y) = y 2 > 0#

(x, 2x2 ), x = 0

(0, 0)

(0, y)

f (x, 2x2 ) = (2x2 − x2 )(2x2 − 3x2 ) = −x4 < 0,

: ! (0, 0) "0 0

x<0: #

6 (0, 0) y = kx, (k ∈ R)

(0, 0)

! f

"0

x ? ' "

y=kx

) f 5

(0, 0) ! ,. -

= f (x, 0) = 4x2 ,

Mmin = (0, 0) :

y=0

1) f 5

y ? ' "

! ! ,. - = f (0, y) = y 2 ,

y = kx, k = 0

%%. f y=kx

Mmin = (0, 0) :

x=0

: f 5

(0, 0)

# W)

= f (x, kx) = (kx − x2 )(kx − 3x2 ) = 3x4 − 4k x3 + k 2 x2 : ?

d f (x, kx) dx

25 ) d2 f (x, kx) dx2

= 0, x=0

x = 0 f (x, kx)

(0, 0) f (x, y)

= 2 k 2 > 0, x=0

# C& * 21 1 def

z = f (x, y) = xy ,

x > 0,

y > 0, '(

? ?# U' ) f ( x2 y2 + = 1 )! ) )! ? ? # :? ! x2 y 2 . ! + − 1 = 0 ' ⎧ ⎪ x, y > 0, ⎨z = f (x, y) = xy −→ max. +3@25. y x ⎪ ⎩ϕ(x, y) def + −1=0: = U

. )

) ) ! (

B ) )!

! '

! ) D '

1) + K) !

%%.(

! ϕ(x, y) = 0 '

' y5 x5 f √ y = y(x) = 12 8 − x2 z = z(x) = f x, y(x) = A . '

1 √ x 8 − x2 ,

0 <

√ 0≤x≤2 2 :

%%. z(x)

. # N) ) !" ) z(x)

%. )

) ? ,'

* 0def

g(x) = x2 (8 − x2 ) −→ max. !" xmax = 2

:

√ 0 ≤ x ≤ 2 2, 0 x2 = 4

" % ' "

g(x)

x = 2 # y

√

r 2 '$

M (2, 1) r 0

√

2 2

&% 9

7) y(2) = 1 " !

f (x, y)

Mmax = M0 (2, 1) ,

(2, 1)

f 5

max f (x, y) = fmax = f (M0 ) = f (2, 1) = 2 :

2)$ + N)! % ! ' ! ⎧ √ ⎨x = x(t) = 2 2 cos t π 0≤t≤ : √ ⎩y = y(t) = 2 sin t U

f (x, y)5 (

. 0 %%. z(t) = f x(t), y(t)

z = z(t) = f x(t), y(t) = 4 cos t sin t = 2 sin 2t −→ max.

0≤t≤

π :

7 0 z(t) π π π tmax = x( 4 ) = 2, y( 4 ) = 1 f (x, y) ! (2, 1) # C" . ! Mmax = M0 (2, 1) ,

max f (x, y) = fmax = f (M0 ) = f (2, 1) = 2 :

3)$ + : !

+3@25. .

) D # $" f 5 D 2 y2 x L(x, y) = f (x, y) + λϕ(x, y) = xy + λ + −1 ,

λ ∈ R,

λ5 D 0" ! #

) D (

)

" ) ) −1

)

# +. D L(x, y)

1 ' " ) ') ' ⎧ ⎧ x ⎧ ⎪ (x, y) = y + λ x = L ⎪ ⎪ λ=− x x=2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y ⎨ ⎨ ⎨ Ly (x, y) = x + λ y = 0 x2 − 4y 2 = 0 ⎪y = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎩ ϕ(x, y) = x + y − 1 = 0 ⎩x2 + 4y 2 = 8 λ = −2 : : ! M0 (2, 1) D L(x, y)

:

! # 7 D L(x, y) ) M0 (2, 1)

1 1 , 2 2

25 (

) λ = −2 Lxx (2, 1) = − , Lxy (2, 1) = 1,

Lyy (x, y) = λ,

Lyy (2, 1) = −2,

1 1 , 2 2

dx +

2Lxy

1 1 , 2 2

Lyy

1 1 , 2 2

dx dy + dy 2 = 1 = − dx2 + 2 dx dy − 2 dy 2 = − dx2 − 4 dx dy + 4 dy 2 = 2 1 +3@27. = − dx − 2 dy : =

Lxx

λ , Lxy (x, y) = 1,

Lxx (x, y) =

7) H 3225 !

+3@25. . ! '

+3@26.

!' M0 (2, 1)

= +3@18.

dϕ(x, y) = ϕx (x, y) dx + ϕy (x, y) dy =

x dx + y dy

dϕ(M0 ) = 0

dϕ(2, 1) = ϕx (2, 1) dx + ϕy (2, 1) dy =

dx + dy = 0 ⇐⇒

dx = −2 dy :

W * ! 25 +3@27. ) d2 L(2, 1) = −

2 2 1 1 dx − 2 dy = − 2 dx = −2 dx2 < 0,

; D

dx = 0 :

25 ) . 0(

d L(2, 1) < 0 0 dx = 0 ' < H 322 25

M0 (2, 1) f 5 ,. - !

M0 (2, 1) ! ϕ(x, y) = 0 ' !

! )# /

Mmax =

!' (

)

Mmax = M0 (2, 1) ,

max f (x, y) = fmax = f (M0 ) = f (2, 1) = 2 :

A%(+. + 32 V !

! ' H 3225# W)

(

' % 0 ' ? √ √ # )! (2 2, 0), (0, 2)

, )! ?

# ; f 5

f (x, y) = xy

f 5

/ 6 9 , 9 C& * 22 + x5 !) y5 . ( # $!)

3000

' . 5000 ' # 1 . (

') 0 <

def

f (x, y) = 120 x4/5 y 1/5 ,

x ≥ 0,

y≥0:

, )-

$ 0

f (K, L)

) 600 '" # ; S

!) ! 0.) ! " ' ) ?) # 7) ?) # D

. # $!)

3000 x . '

5000 y # J ! ' '"

: ! '

3000 x + 5000 y = 600 000

600

3 x + 5 y = 600,

? ! . ⎧ ⎨z = f (x, y) = 120 x4/5 y 1/5 −→ max. x, y ≥ 0, ⎩ϕ(x, y) def = 3 x + 5 y − 600 = 0 : X ) D " f 5 D

+3@28. (

L(x, y) = f (x, y) + λϕ(x, y) = 120 x4/5 y 1/5 − λ(3 x + 5 y − 600),

λ ∈ R,

λ5 D 0" ! ,λ5 -# E D L(x, y)

# 1

' " ) ') ' ⎧ ⎧ ⎪ Lx (x, y) = 120 x−1/5 y 1/5 − 3 λ = 0 ⎪ 96 x−1/5 y 1/5 = 3 λ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ 24 x4/5 y −4/5 = 5 λ Ly (x, y) = 120 x4/5 y −4/5 − 5 λ = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ 3 x + 5 y = 600, ϕ(x, y) = 3 x + 5 y − 600 = 0 ⎧ ⎧ ⎧ ⎨96 · 5 x−1/5 y 1/5 = 24 · 3 x4/5 y −4/5 ⎨y = 0, 15 x ⎨x = 160 ⎩3 x + 5 y = 600 ⎩3 x + 5 y = 600 ⎩y = 24 :

: ! M0 (160, 24) D L(x, y)

+3@28. . ! '

D L(x, y)

0 ! ) ) 25 )

! . B

A ) B

. %) ) ' B) (

3 x+5 y = 600

+3@28. . 15 ?

' # : '

' '

" '

f (200, 0) = 0,

f (0, 120) = 0,

f (x, y) > 0,

3 x + 5 y = 600,

x, y > 0 :

70 ) ' M0 (160, 24) f

) )

Mmax = M0 (160, 24), fmax = f (M0 ) = f (160, 24) = 120 (160)4/5 (24)1/5 ≈ 13 138 : : ! !

0. 160

160 · 3000 = 480 000 - 24 . ,B 24 ' . "

24 · 5000 = 120 000 - ! "

' ?) # : ?) " ! 13 138 #

* 9:

L&/ &:

5%5% * 6+*7 & ; , & ''( ( ,! !

(

!, % '( ;N 9 N% , # ( 6 ''(

( % "

! 6

" ''(

" % C& * 23 1 def

u = f (M ) = f (x, y, z) = x + y + z 2 : R3 −→ R 0 R3 ?%

! ⎧ ⎪ u = f (x, y, z) = x + y + z 2 −→ extr. ⎪ ⎪ ⎨ def ϕ1 (x, y, z) = z − x − 1 = 0 ⎪ ⎪ ⎪ ⎩ϕ (x, y.z) def = y − xz − 1 = 0 :

! '(

+3@29.

+ ! ϕ1 = 0 ϕ2 = 0 '

5 Γ ⊂ R

Γ f 5

'"(

f (M )

M ∈Γ

* f # Γ

$! ' x %%. ⎧ ⎧ ⎧ ⎨z = x + 1 ⎨z = x + 1 ⎨z − x − 1 = 0 ⎩ y = x2 + x + 1 : ⎩y = xz + 1 ⎩y − xz − 1 = 0

W) f

0 <

%%.(

u = u(x) = f (x, x2 + x + 1, x + 1) = x + x2 + x + 1 + (x + 1)2 = = 2x2 + 4x + 2 = 2(x + 1)2 −→ extr. A

xmin = −1 , umin = 0 # V<(

fmin = 0

!" . # :?

"0 f

x∈R:

Mmin = M0 (−1, 1, 0)

C& * 24 1 def

u = f (x, y, z) = x2 − y 2 + z 2 : R3 −→ R 0 R3 ?% ! ⎧ ⎨u = f (x, y, z) = x2 − y 2 + z 2 −→ extr.

+3@30.

⎩ϕ(x, y, z) def = 2x − y − 3 = 0 : + ! ϕ = 0 ' ' '"

Γ ⊂ R

'"(

W) ! y = 2x − 3 '

y %%.

0 <

u = u(x, z) = f (x, 2x − 3, z) = x2 − (2x − 3)2 + z 2 = = −3x2 + 12x − 9 + z 2 −→ extr.

x, z ∈ R,

))

u(x, z)

⎧ ∂u ⎪ ⎨ = −6x + 12 = 0 ∂x ⎪ ⎩ ∂u = 2z = 0 ∂z

x = 2, z = 0

u(x, z)

f (x, y, z) N

⎧ ⎨x = 2 ⎩z = 0 : ' ,))-

: ' ! .

0 ! )

; ) H 1275 "

) 25

H 127-

−6 0

= −12 < 0 :

Δ 0

x = 2, z = 0

M0 (2, 1, 0) (

' #

Δ ') x = 2, z = 0 ⎧ 2 ∂ u ⎪ ⎪ ⎪ 2 = −6 ⎪ ⎪ ∂x ⎪ ⎨ 2 uxx uxz ∂ u Δ= = ⎪ ∂z 2 uzx uzz ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ u =0 ∂x∂z

u(x, z)

' )# +.

# V<!) M0 (2, 1, 0) !

?' !

f (x, y, z)

S , 6

''( " % " , 9

" %

$ ,$ '#

'( ) $

, ''(

$& < 6+*7

64 #

$& < ,.%L/

+ ! .

" 9$ ( #

⎧ u = f (x, y, z) −→ extr. ⎪ ⎪ ⎨ ϕ1 (x, y, z) = 0 ⎪ ⎪ ⎩ ϕ2 (x, y.z) = 0 :

; ϕ1 = 0 ϕ2 = 0

> Γ ⊂ R

% '

Γ f > "$ %&

f P 23> ; % '

, > S ⊂ R3

f ) $ !

& ! "$ %& f

f ) $ !

& ! Γ

+3@31.

S f >

P 24> +3@31. ( 9 4 % %

9 f ) $ $& < 6+*7 & def

L(x, y, z) ≡ L(x, y, z, λ1 , λ2 ) = f (x, y, z) + λ1 ϕ1 (x, y, z) + λ2 ϕ2 (x, y, z) , +3@32. " (x, y, z) ∈ Ω % ''( λ1 , λ2 ∈ R - *Q&

+ . Ω ⊂ R

$& < ! P, @

$& < %&%) $%&' *7& S ''#

( N

'%&

f (x, y, z) 6+*7

* 9:

L&/ ; , &<: 9

f R $& < 6+*7

* 4%* 4 9: TU

& f (x, y, z) ) $

L&=

( 4

% 4 f > 9 4 ) $ $

9 9 & ⎧ ∂ϕ1 ∂ϕ2 ∂f ⎪ ⎪ + λ1 + λ2 =0 ⎪ ⎪ ∂x ∂x ∂x ⎪ ⎪ ⎪ ⎪ ∂ϕ1 ∂ϕ2 ∂f ⎪ ⎪ + λ1 + λ2 =0 ⎪ ⎪ ∂y ∂y ⎨ ∂y ∂ϕ1 ∂ϕ2 ∂f ⎪ + λ1 + λ2 =0 ⎪ ⎪ ∂z ∂z ∂z ⎪ ⎪ ⎪ ⎪ ⎪ϕ1 (x, y, z) = 0 ⎪ ⎪ ⎪ ⎪ ⎩ϕ (x, y, z) = 0

⎧ ⎪ Lx = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L =0 ⎪ ⎪ ⎨ y Lz = 0 ⎪ ⎪ ⎪ ⎪ L = 0 ⎪ λ1 ⎪ ⎪ ⎪ ⎪ ⎩ L = 0 : λ2

⇐⇒

A 9% 9 +3@33. 9

+3@33.

,"$ 9

% f ) $

%&, 33 ,U

' ! -

1 f (x, y, z), ϕ1 (x, y, z), ϕ2 (x, y, z)

Ω ⊂ R +3@31. . )

f, ϕ1 , ϕ2 ∈ C (Ω) M0 (x0 , y0 , z0 ) ∈ Ω

f 5 !

ϕ1 = 0 ϕ2 = 0 ! '

& M0 (x0 , y0 , z0 )

ϕ 1 ϕ2

) " ') ,Z0- ⎛

∂ϕ1 (M0 ) def ⎜ ∂x J = J(M0 ) = ⎝ ∂ϕ2 (M0 ) ∂x " ? ' : !

⎞ ∂ϕ1 (M0 ) ⎟ ∂z ∂ϕ2 (M0 ) ⎠ ∂z

# :) ! J

25 rank J = 2#

λ1 , λ2 ∈ R !

(x0 , y0 , z0 , λ1 , λ2 ) ' 0 D L

∂ϕ1 (M0 ) ∂y ∂ϕ2 (M0 ) ∂y

+3@33. '

= 4 -

''( $ ( ) $ %&, 34 ,U

0 ! -

1 25 u = f (M ) = f (x, y, z)

. ! ϕ1 (x, y, z) = 0 ϕ2 (x, y.z) = 0

! ⎧ u = f (x, y, z) −→ extr. ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ϕ1 (x, y, z) = 0

+3@34.

ϕ2 (x, y.z) = 0 :

1 M0 (x0 , y0 , z0 ) 5 λ1 , λ2 !

f (x, y, z)

D L(x, y, z) ≡ L(x, y, z, λ1 , λ2 ) = f (x, y, z) + λ1 ϕ1 (x, y, z) + λ2 ϕ2 (x, y, z) ,

dL(M0 ) ≡ dL(x0 , y0 , z0 ) ≡ 0

⎧ ∂ϕ1 ∂ϕ2 ∂f ⎪ ⎪ (M0 ) + λ1 (M0 ) + λ2 (M0 ) = 0 ⎪ ⎪ ∂x ∂x ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂f (M0 ) + λ1 ∂ϕ1 (M0 ) + λ2 ∂ϕ2 (M0 ) = 0 ⎪ ⎪ ∂y ∂y ⎨ ∂y ∂ϕ1 ∂ϕ2 ∂f ⎪ (M0 ) + λ1 (M0 ) + λ2 (M0 ) = 0 ⎪ ⎪ ∂z ∂z ∂z ⎪ ⎪ ⎪ ⎪ ⎪ ϕ1 (M0 ) = 0 ⎪ ⎪ ⎪ ⎪ ⎩ϕ (M ) = 0 :

$" D

+3@35.

25 ) M0 (x0 , y0 , z0 )

d2 L(M0 ) = Lxx (M0 ) dx2 + +Lyy (M0 ) dy 2 + Lzz (M0 ) dz 2 + + 2Lxy (M0 ) dx dy + 2Lxz (M0 ) dx dz + 2Lyz (M0 ) dy dz : +3@36.

1? C d2 L(M0 ) > 0 0) ∀ dx, dy, dz ) ,*'

dx2 + dy 2 + dz 2 = 0

! ⎧ ⎧ ⎪ ∂ϕ1 (M ) dx + ∂ϕ1 (M ) dy + ∂ϕ1 (M ) dz = 0 ⎨dϕ1 (M0 ) = 0 ⎪ ⎨ ∂x ∂y ∂z ∂ϕ2 ∂ϕ2 ⎩dϕ2 (M0 ) = 0 ⎪ ⎪ ∂ϕ2 ⎩ (M0 ) dx + (M0 ) dy + (M0 ) dz = 0, ∂x ∂y ∂z +3@37.

! M0 (x0 , y0 , z0 ) f

,. - !

(

M0 (x0 , y0 , z0 ) = Mmin ! ϕ1 = 0 ϕ2 = 0

2? C d2 L(M0 ) < 0 0) ∀ dx, dy, dz ) ,*' 0

! +3@37. '

! M0 (x0 , y0 , z0 ) f

dx2 + dy 2 + dz 2 = 0

,. - !

M0 (x0 , y0 , z0 ) = Mmax ! ϕ1 = 0 ϕ2 = 0

3? C d2 L(M0 ) ) %. , ( 0 - dx, dy, dz ) ,*M0 (x0 , y0 , z0 ) f

+3@37. ' %

(

# 4? C d2 L(M0 ) ≥ 0

d2 L(M0 ) ≤ 0 0 dx, dy, dz )(

,*- 0

+3@37. ' %

! ) 2

) ) ! f 5 !

! ) ) ) ! ,

(

M0 (x0 , y0 , z0 )

(

!' )

= & 4

+B V

1@6.

9 ) $%

+9 4 ) $ . $

% W

6 $

,! 9 4

C& * 25 1 K 235

def

u = f (M ) = f (x, y, z) = x + y + z 2 : R3 −→ R

! ' ! ⎧ ⎪ u = f (x, y, z) = x + y + z 2 −→ extr. ⎪ ⎪ ⎨ def ϕ1 (x, y, z) = z − x − 1 = 0 ⎪ ⎪ ⎪ ⎩ϕ (x, y.z) def = y − xz − 1 = 0 :

+3@38.

. K 235

%%.

)

+3@38. . ) D )

f 5 D

%%.

' # $"(

L ≡ L(x, y, z, λ1 , λ2 ) = f (x, y, z) + λ1 ϕ1 (x, y, z) + λ2 ϕ2 (x, y, z) = = x + y + z 2 + λ1 (z − x − 1) + λ2 (y − xz − 1) ) ) +3@33. ⎧ ⎧ ⎪ ⎪ L = − λ − λ z = λ1 = z + 1 ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ = −1 L = 1 + λ2 = 0 ⎪ ⎪ ⎪ ⎪ ⎨ y ⎨ 2 Lz = 2z + λ1 − λ2 x = 0 x + 3z = −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z−x=1 ϕ1 = z − x − 1 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ϕ = y − xz − 1 = 0 ⎩y − xz = 1

' ⎧ ⎪ λ1 = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ = −1 ⎪ ⎪ ⎨ 2 x = −1 ⎪ ⎪ ⎪ ⎪ ⎪ y=1 ⎪ ⎪ ⎪ ⎪ ⎩z = 0 :

: ! M0 (−1, 1, 0) D

f (M )

' # N

0 ! )

' D(

25 ) Lxx = 0,

Lxy = 0,

Lyy = 0,

Lxz = −λ2 = 1,

Lzz = 2,

Lyz = 0,

d2 L(M0 ) = Lxx (M0 ) dx2 + +Lyy (M0 ) dy 2 + Lzz (M0 ) dz 2 + + 2Lxy (M0 ) dx dy + 2Lxz (M0 ) dx dz + 2Lyz (M0 ) dy dz = = 2 dz 2 + 2 dx dz = 2 dz dz + dx : & !' M0 +3@37. ' ⎧ ∂ϕ1 ∂ϕ1 ⎪ ⎪ dx + dy + ⎨ ∂x ∂y ∂ϕ2 ∂ϕ ⎪ ⎪ ⎩ 2 dx + dy + ∂x ∂y

∂ϕ1 dz = 0 ∂z ∂ϕ2 dz = 0 ∂z

⎧ ⎨dx − dz = 0 ⎩z dx − dy + x dz = 0

⎧ ⎨dx − dz = 0 ⎩ − dy − dz = 0

dx = dz = −dy # W * ! D

25 ) d2 L(M0 ) = 2 dz dz+dx = 2 dz dz+dz = 4 dz 2 > 0, ; d2 L(M0 ) > 0 0 dx, dy, dz *

dx + dy + dz = 0 0 0, dϕ2 = 0 '

dz = 0 :

dϕ1 =

! ' < H 324 15 M0

(

fmin = 0 #

Mmin = M0 (−1, 1, 0)

C& * 26 1 def

u = f (x, y, z) = x2 − y 2 + z 2 : R3 −→ R

'") K 245

! ' ⎧ ⎨u = f (x, y, z) = x2 − y 2 + z 2 −→ extr. ⎩ϕ(x, y, z) def = 2x − y − 3 = 0 :

W

+3@39.

W %. ? D # $" f 5 D

L ≡ L(x, y, z, λ) = f (x, y, z) + λϕ(x, y, z) = x2 − y 2 + z 2 + λ(2x − y − 3) ) ) +3@33. ' ⎧ ⎧ ⎧ ⎪ ⎪ ⎪ = 2x + 2λ = x + λ = L λ = −2 ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨λ = −2y ⎨L = −2y − λ = 0 ⎨x = 2 y ⎪z = 0 ⎪L = 2z = 0 ⎪ ⎪ ⎪ ⎪ y=1 z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩2x − y = 3 ⎩ϕ = 2x − y − 3 = 0 ⎩z = 0 : : ! M0 (2, 1, 0) D

f (M )

' # N

0 ! )

' D(

25 ) Lxx = 2,

Lxy = 0,

Lyy = −2,

Lxz = 0,

Lzz = 2,

Lyz = 0,

d2 L(M0 ) = Lxx (M0 ) dx2 + Lyy (M0 ) dy 2 + Lzz (M0 ) dz 2 +

+ 2Lxy (M0 ) dx dy + 2Lxz (M0 ) dx dz + 2Lyz (M0 ) dy dz = = 2 dx2 − 2 dy 2 + 2 dz 2 : & !' M0

∂ϕ ∂ϕ ∂ϕ dx + dy + dz = 0 ∂x ∂y ∂z

⇐⇒

2 dx − dy = 0 :

W * dy = 2 dx ! D

) d2 L(M0 ) = 2 dx2 − dy 2 + dz 2 = 2 dz 2 − 3 dx2 : 7

) d2 L(M0 ) )

) 0

B d2 L(M0 ) = −4 dx2 < 0,

d L(M0 ) = 2 dx > 0,

dz = dx = 0,

dz = 2 dx = 0 :

; d2 L(M0 ) ) %. 0 dx, dy, dz * " dx2 + dy 2 + dz 2 = 0 0

dϕ = 0 ' ! ' < H 324 35 M0 (2, 1, 0)

(

?' !

(

C& * 27 7"

def

u = f (M ) = f (x, y, z) = x2 + y 2 + z 2 : R3 −→ R+ x2 y 2 z 2 + + = 1 ' ! a2 b 2 c 2 ,0 < a < b < c-# C% '") f

0 z2 =1 c2

)

x2 y2 + + a2 b2

⎧ ⎪ ⎨u = f (x, y, z) = x2 + y 2 + z 2 −→ extr.

+3@40. x2 y 2 z 2 ⎪ ⎩ϕ(x, y, z) def = 2 + 2 + 2 −1=0: a b c $" f D 2 x y2 z2 + + − L(x, y, z, λ) = f (x, y, z) + λϕ(x, y, z) = x2 + y 2 + z 2 + λ a2 b2 c2 ) ) +3@33. ' ⎧ ⎧ λ ⎪ 2λ x ⎪ x 1+ 2 =0 ⎪ ⎪ ⎪ ⎪ Lx = 2x + 2 = 0 a ⎪ ⎪ ⎪ ⎪ a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L = 2y + 2λ y = 0 ⎪y 1 + λ = 0 ⎨ ⎨ y b2 b2 2λ z ⎪ ⎪ λ ⎪ ⎪ Lz = 2z + 2 = 0 ⎪ ⎪ =0 z + ⎪ ⎪ ⎪ ⎪ c ⎪ ⎪ c2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ L ≡ ϕ = x + y + z − 1 = 0 ⎪ ⎪ ⎩x + y + z = 1 : λ a b c a b c P%. ' % 0 < a < b < c ! ' ? ) ) ' 0) )

0 0 0

λ = −a2 ,

M = M3,4 0, ±b, 0 ,

M = M5,6 0, 0, ±c :

λ = −b , λ = −c ,

M = M1,2 ± a, 0, 0 ,

A D f (M )

# 1

' #

0 ! )

' D(

25 ) (

Lxx = 2 +

2λ , a2

Lxy = 0,

2λ , b2 2λ =2+ 2, c

Lyy = 2 +

Lxz = 0,

Lzz

Lyz = 0,

d2 L(M, λ) = Lxx (M ) dx2 + Lyy (M ) dy 2 + Lzz (M ) dz 2 + + 2Lxy (M ) dx dy + 2Lxz (M ) dx dz + 2Lyz (M ) dy dz = λ λ λ = 2 1 + 2 dx + 1 + 2 dy + 1 + 2 dz : a b c 25 d2 L(M, λ) (

+ D ) x, y, z %%.

(

! )

λ5 ' ! . # & !' Mi (i = 1, 6)

! dϕ = 0 '(

∂ϕ ∂ϕ ∂ϕ dx+ dy+ dz = 0 ∂x ∂y ∂z :? <

⇐⇒

Mi (i = 1, 6)

d L(M, λ) ) (

" # 6" '

x y z dx+ 2 dy+ 2 dz = 0 : +3@41. a2 b c

M = M1,2 ± a, 0, 0 a2 a2 d2 L M1,2 , −a2 = 2 1 − 2 dy 2 + 1 − 2 dz 2 > 0, b c

C0 λ = −a

0 < a < b < c# 70 H 324 15 Mmin = M1,2 ± a, 0, 0 6

f (M )5 !

) fmin = f (M1,2 ) = a #

C0 λ = −b2 M = M3,4 0, ±b, 0 b2 b2 d L M3,4 , −b = 2 1 − 2 dx + 1 − 2 dz : a c

7 ? )

d L M3,4 , −b

! )

0 < a < b < c

)

) %. b2 dx = 0, dz = 0, d L M3,4 , −b = 2 1 − 2 dz 2 > 0, c b2 d2 L M3,4 , −b2 = 2 1 − 2 dx2 < 0, dz = 0, dx = 0 : a

70 H 324 35 M3,4 0, ±b, 0 !

f (M )

(

M = M5,6 0, 0, ±c c c d L M5,6 , −c = 2 1 − 2 dx + 1 − 2 dy < 0, a b

C0 λ = −c

0 < a < b < c# 70 H 324 25 Mmax = M5,6 0, 0, ±c 6

f (M )5 !

)

fmax = f (M5,6 ) = c

X ) # 7 ' B ) ! dϕ = 0 +3@41. ' A%(+. + 33

H 3245

25 d L(M ) )

d L(M ) < !'!

) f (M )

! ' '

& 2235

0 !

)

' ? )# ;( '

)

* 9:

L& 4 )

5%5% * 6+*7 & ; , & $ ''( ,! !

(

% !, %

'( ;N 9 N% , ( 6 n ''( $ ( ) $ ( % "

(n−1) %

& m (1 ≤ m ≤ n − 1) =

% " 9$ ( !

( % - $

⎧ ⎨u = f (M ) = f (x1 , x2 , . . . , xn ) − → extr. ⎩ϕi (M ) = 0 (1 ≤ i ≤ m) :

+3@42.

$& < ,.%L/ 64 @

$& < 6+*7 L ≡ L(M, λi ) ≡ L(x1 , x2 , . . . , xn , λ1 , λ2 , . . . , λm ),

M ∈ Ω ⊂ Rn ,

def

L = f (M ) + λ1 ϕ1 (M ) + λ2 ϕ2 (M ) + · · · + λm ϕm (M ) , " λi ∈ R (1 ≤ i ≤ m) - *Q&

$& < ! P, @

$& < %&%) $%&' *7& * 9 4

λi ,! % % ' ( ''(

$ & m < n S

''(

f (M ) 6+*7

* 9:

'%& L&/ ; , &<: 9

f R $& < 6+*7

* 4%* 4 9: TU

+3@43.

& f (M ) ) $

L&=

( 4

% 4 f > 9 4 ) $ $ # 9 9 & ⎧ ∂ϕ1 ∂ϕm ∂f ⎪ ⎨ + λ1 + · · · + λm =0 ∂xj ∂xj ∂xj ⇐⇒ ⎪ ⎩ϕ (x, y, z) = 0 i

⎧ ∂L ⎪ ⎪ = 0 (1 ≤ j ≤ n) ⎨ ∂xj ⎪ ∂L ⎪ ⎩ = 0 (1 ≤ i ≤ m) : ∂λi +3@44.

A 9% n+m $ ,"$ n+m 9 +3@44. 9

% f ) $

%&, 35 ,U

' ! -

1 f (M ) = f (x1 , x2 , . . . , xn ), ϕi (M ) = ϕi (x1 , x2 , . . . , xn ) m < n

f, ϕi ∈ C 1 (Ω)

Ω ⊂ Rn

M0 (x01 , x02 , . . . , x0n ) ∈ Ω ⎧ ⎨u = f (M ) = f (x1 , x2 , . . . , xn ) −→ extr. ⎩ϕi (M ) = 0 (1 ≤ i ≤ m) . )

f 5 !

! '

& M0

ϕi

1≤i≤

+3@45. ϕi = 0

)

" ') ,Z0- ,m × n % ∂ϕi (M0 ) def J = J(M0 ) = ∂xj 1≤i≤m 1≤j≤n

" m5 # :) ! J ? ' : !

m5 rank J = m#

λi ∈ R (1 ≤ i ≤ m) !(

(x1 , x2 , . . . , xn , λ1 , λ2 , . . . , λm ) ∈ Rn+m 0 ⎧ ⎧ ∂L ⎪ ∂ϕ ∂ϕ ∂f m ⎪ ⎪ = 0 (1 ≤ j ≤ n) ⎨ ⎨ + λ1 + · · · + λm =0 ∂xj ∂xj ∂xj ∂xj ⇐⇒ ⎪ ⎪ ∂L ⎪ ⎩ϕ (x, y, z) = 0 ⎩ = 0 (1 ≤ i ≤ m) : i ∂λi +3@46. ' D L

# = % Q V %

N - @ f

) $

% f > 9 4 ) $

$ = 4 -

''( $ ( ) $ %&, 36 ,U

0 !

1 25 u = f (M ) = f (x1 , x2 , . . . , xn ) !

. ! ϕi = 0 '

⎧ ⎨u = f (M ) = f (x1 , x2 , . . . , xn ) − → extr. ⎩ϕi (M ) = 0 (1 ≤ i ≤ m) :

+3@47.

1 M0 (x01 , x02 , . . . , x0n ) 5 λi (1 ≤ i ≤ m) ! D

f (M )

L ≡ L(M ) ≡ L(M, λi ) = f (M ) + λ1 ϕ1 (M ) + · · · + λm ϕm (M ) dL(M0 ) ≡ 0 ⎧ ⎪ ⎨ ∂f (M0 ) + λ1 ∂ϕ1 (M0 ) + · · · + λm ∂ϕm (M0 ) = 0 (1 ≤ j ≤ n) ∂xj ∂xj ∂xj ⎪ ⎩ϕ (M ) = 0 (1 ≤ i ≤ m) : i

$" D d2 L(M0 ) =

25 ) M0

∂ 2 L(M0 ) 2 ∂ 2 L(M0 ) 2 ∂ 2 L(M0 ) dxj dxk = dx1 + · · · + dxn + ∂xj ∂xk ∂x1 ∂x2n j,k=1 +2

1? C

+3@48.

∂ 2 L(M0 ) ∂ 2 L(M0 ) dx1 dx2 + · · · + 2 dxn−1 dxn : ∂x1 ∂x2 ∂xn−1 ∂xn

d2 L(M0 ) > 0

∀ dxj (1 ≤ j ≤ n)

)

,*- ' " dx21 +· · ·+dx2n = 0 0

! dϕi (M0 ) = 0

(1 ≤ i ≤ m),

+3@49.

! M0 f

,. - !

M0 = Mmin ! ϕi = 0 ' d2 L(M0 ) < 0

2? C

∀ dxj (1 ≤ j ≤ n)

)

,*- ' " dx21 +· · ·+dx2n = 0 0

! dϕi (M0 ) = 0

! M0 f

(1 ≤ i ≤ m),

,. - !

M0 = Mmax ! ϕi = 0 '

+3@50.

3? C d2 L(M0 ) ) %. , ( 0 - dxj )

+3@50. ' %

4? C d2 L(M0 ) ≥ 0 ,*- 0

! M0 #

d2 L(M0 ) ≤ 0 0 dxj )

+3@50. ' %

) ' 2 M0 !

f 5 ! W

) ) ! ) ) ) ,

!' ) '"

n ''( $ ( ) $ 6

% ,!

9 4 C& * 28 D !

') .

! ! ' ⎧ ⎨u = f (M ) = f (x1 , x2 , . . . , xn ) def = x1 + x2 + · · · + xn −→ extr. ⎩ϕ(M ) def = x21 + x22 + · · · + x2n − 1 = 0 :

+3@51.

! ) R

def S = M (x1 , x2 , . . . , xn ) ∈ Rn : x21 + x22 + · · · + x2n = 1

# : !'

'") f

# $" f 5 D

(

λ 0"(

L ≡ L(M ) ≡ L(x1 , x2 , . . . , xn , λ) = f (M ) + λϕ(M ) = = x1 + x2 + · · · + xn + λ(x21 + x22 + · · · + x2n − 1) A"0

,' (

-# : ! " ) ')

' ⎧ ⎧ ∂L ⎪ ⎪ = 1 + 2λ xj = 0 (1 ≤ j ≤ n) ⎨ ⎨xj = −1 (1 ≤ j ≤ n) ∂xj 2λ ⇐⇒ ⎪ ⎩ 2 ∂L ⎪ x1 + x22 + · · · + x2n = 1 ⎩ =ϕ=0 ∂λ ⎧ √ ⎧ n −1 ⎪ ⎪ ⎪ ⎨λ = ± ⎨ xj = (1 ≤ j ≤ n) 2λ ⇐⇒ ⇐⇒ ⎪ ⎪ ⎪ ⎩n 1 = 1 (1 ≤ j ≤ n) : ⎩ xj = ∓ √ n 4λ A ,S - √ −1 −1 n 0 λ = ∈ S, , M = M1 √ , . . . , √ n n √ n 0 λ = − ∈S: , M = M2 √ , . . . , √ n n N

0 ! )

' D

25 ) ) ∂ 2L = 2λ ∂x2j

∂ 2L =0 ∂xj ∂xk

(1 ≤ j ≤ n),

(j = k),

d2 L(M ) =

n ∂ 2L 2 ∂ 2L ∂ 2 L(M ) dxj dxk = dx1 + · · · + 2 dx2n + ∂xj ∂xk ∂x1 ∂xn j,k=1

∂ 2L ∂ 2L dx1 dx2 + · · · + 2 dxn−1 dxn = +2 ∂x1 ∂x2 ∂xn−1 ∂xn = 2λ dx21 + · · · + dx2n : 7 < H 3265 !' ! dϕ(M1,2 ) =

∂ϕ(M1,2 ) ∂ϕ(M1,2 ) dx1 + · · · + dxn = 0, ∂x1 ∂xn −1 −1 dx1 + · · · + dxn = 0, 2 x1 dx1 + · · · + 2 xn dxn = λ λ dx1 + · · · + dxn = 0 :

R d2 L ) !"

") d L ) (

# C0 M = M1 , λ = D f fmin

√

C0 M = M2 , λ = − f

d2 L(M1 ) > 0

))

! √ = f (M1 ) = − n#

! (

√

d2 L(M2 ) < 0

C& * 29 D !

') . (

! ' ⎧ ⎨u = f (M ) = f (x1 , x2 , . . . , xn ) def = x21 + x22 + · · · + x2n −→ extr. ⎩ϕ(M ) def = x1 + x2 + · · · + xn − 1 = 0 :

))

! √ fmax = f (M2 ) = n#

f ?

) R

x1 + x2 + · · · +

xn − 1 = 0 '!' # $" f 5 D

λ 0" (

L ≡ L(M ) ≡ L(x1 , x2 , . . . , xn , λ) = f (M ) + λϕ(M ) = = x21 + x22 + · · · + x2n + λ(x1 + x2 + · · · + xn − 1) A"0

,' (

-# : ! " ) ')

' ⎧ ∂L ⎪ ⎪ = 2 xj + λ = 0 (1 ≤ j ≤ n) ⎨ ∂xj ⇐⇒ ⎪ ∂L ⎪ ⎩ =ϕ=0 ∂λ ⎧ −λ ⎪ ⎨ xj = (1 ≤ j ≤ n) ⇐⇒ ⇐⇒ ⎪ ⎩ −λ n = 1 A 0 N

−2 λ= , n

⎧ ⎨xj = −λ (1 ≤ j ≤ n) ⎩ x1 + x2 + · · · + x n = 1 ⎧ −2 ⎪ ⎨λ = n ⎪ ⎩xj = 1 (1 ≤ j ≤ n) : n

M = M0

0 ! )

n n

' D

25 ) ) ∂ 2L =2 ∂x2j

d2 L(M ) =

∂ 2L =0 ∂xj ∂xk

(1 ≤ j ≤ n),

(j = k),

n ∂ 2L 2 ∂ 2L ∂ 2 L(M ) dxj dxk = dx1 + · · · + 2 dx2n + ∂xj ∂xk ∂x1 ∂xn j,k=1

∂ 2L ∂ 2L +2 dx1 dx2 + · · · + 2 dxn−1 dxn = ∂x1 ∂x2 ∂xn−1 ∂xn = 2 dx21 + · · · + dx2n > 0, 0 dx21 + · · · + dx2n = 0 : 7 < H 3265 M0 D

' ? )

fmin = f (M0 ) =

n

! dϕ(M0 ) = 0 ' (

)) (

)

C& * 30 D !

! ' ⎧ n ⎪ def ⎪ ⎨u = f (M ) = ak x2k −→ extr.

') . (

(0 < a1 < a2 < · · · < an )

k=1

⎪ ⎪ ⎩ϕ(M ) def = x21 + x22 + · · · + x2n − 1 = 0 : C% def

S =

f ? ) Rn

M (x1 , x2 , . . . , xn ) ∈ Rn :

x21 + x22 + · · · + x2n = 1

# : !'

" ! ,.

'") f

# $" f 5 D

(

λ 0(

λ5 %.

−λ- L ≡ L(M ) ≡ L(x1 , x2 , . . . , xn , λ) = f (M ) − λϕ(M ) = n n ak x2k − λ(x21 + x22 + · · · + x2n − 1) = (ak − λ) x2k + λ : = k=1

A"0

k=1

,' (

-# : ! " ) ')

' ⎧ ∂L ⎪ ⎪ = 2(aj − λ) xj = 0 ⎨ ∂xj ⎪ ⎪ ⎩ ∂L = ϕ = 0 ∂λ

(1 ≤ j ≤ n)

U" ' ak

⇐⇒

⎧ ⎨ λ = aj

xj = 0 ⎩ x2 + x2 + · · · + x2 = 1 : n

' ! 0 λ5 '

λ = aj = ak (j = k)

' a1 < a2 < · · · < an ! 1, n) xj = 0

λ = aj (j =

0) j (j = 1, n) '

! ϕ = 0 '

M (x1 , . . . , xn )

! ) S # 70

λ = ak

k ∈ [1, n] ' # :

' 0 λ = aj xj = 0

0) j (j = k)

' ' ' ')

∃k∈

[1, n] ! ⎧ λ = ak ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

xj = 0

⎧ λ = ak ⎪ ⎪ ⎨

∀j=k

⎪ ⎪ ⎩

x2k = 1

xj = 0

∀j=k

xk = ±1 :

: ! 2n ' ,0 0 0) S - 0

λ = ak ,

M = Mk (0, . . . , 0, ±1, 0, . . . , 0) ∈ S,

k = 1, 2, . . . , n,

1 −1 Mk k5 + S !

- R n

f ∈ C(S) '

' % 0"

(

' < V

# C . f

% )

a1 < a2 < · · · < an ! " )

max f (M ) = f Mn(±) = an : M ∈S

Mmin = M1 (±1, 0, . . . , 0) ∈ S,

6 Mk

Mmax = Mn(±) (0, . . . , 0, ±1) ∈ S, + !

min f (M ) = f M1

M ∈S

(1 < k < n) ?' f

# : % )

H 3265 D

= a1 :

) ) ∂ 2L = 2(aj − λ) ∂x2j

(1 ≤ j ≤ n),

∂ 2L =0 ∂xj ∂xk

(j = k),

n ∂ 2L 2 ∂ 2L 2 ∂ 2 L(M ) d L(M ) = dxj dxk = dx + · · · + dx + ∂xj ∂xk ∂x21 ∂x2n n j,k=1

∂ 2L ∂ 2L dx1 dx2 + · · · + 2 dxn−1 dxn = ∂x1 ∂x2 ∂xn−1 ∂xn

= 2(a1 − λ) dx21 + · · · + 2(ak − λ) dx2k + · · · + 2(an − λ) dx2n :

C0 M = Mk (0, . . . , 0, ±1, 0, . . . , 0) λ 0" !

(

λ = ak '0

d 2 L Mk

=2(a1 − ak ) dx21 + · · · + 2(ak−1 − ak ) dx2k−1 + + 2(ak+1 − ak ) dx2k+1 + · · · + 2(an − ak ) dx2n :

+ M = Mk (0, . . . , 0, ±1, 0, . . . , 0) ?

d L

+3@52.

)

! %. dϕ(Mk ) =

∂ϕ(Mk ) ∂ϕ(Mk ) dx1 + · · · + dxn = 0, ∂x1 ∂xn 2 x1 dx1 + · · · + 2 xn dxn = 0

⇐⇒

!") d2 L Mk

# &

dxk = 0 :

+3@52. )

) +3@52. ) ? k−1 (

0 n−k aj − ak < 0, 1 ≤ j ≤ k − 1, A

aj − ak > 0, k + 1 ≤ j ≤ n :

+3@52. )

= =

0 0 dx1 , dx2 , . . . , dxn , dx21 + · · · + dx2n = 0, * ' #

7 < H 326 35 Mk

f S

(1 < k < n)

! (

A 6

% - #

C& * 31 E ')

% (

⎧ n ⎪ def ⎪ ⎨u = f (M ) = f (x) = aij xi xj −→ max.min. ⎪ ⎪ ⎩

(aij = aji ∈ R)

i,j=1 def

ϕ(M ) = ϕ(x) = x21 + x22 + · · · + x2n − 1 = 0 :

? < ) ) & 2235

C% . (

f ? < %(

! !'

) Rn S '") f

$" f 5 D ,.

λ 0" !

λ5 %.

−λ- L ≡ L(x) ≡ L(x1 , x2 , . . . , xn , λ) = f (x) − λϕ(x) = n aij xi xj − λ(x21 + x22 + · · · + x2n − 1) : = k=1

A"0

,' (

-# : ! " n ' (

0 ') ' ⎧ 1 ∂L ⎪ ⎪ ⎪ ⎪ 2 ∂x1 ⎪ ⎪ ⎪ ⎪ 1 ∂L ⎪ ⎨ 2 ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ∂L 2 ∂xn

= (a11 − λ) x1 + a12 x2 + · · · + a1n xn = 0 = a21 x1 + (a22 − λ) x2 + a23 x3 + · · · + a2n xn = 0 .. .

+3@53.

.. .

= an1 x1 + an2 x2 + · · · + an,n−1 xn−1 + (ann − λ) xn = 0,

) (n + 1)5 ,!- '

ϕ(x) = x21 + x22 + · · · + x2n − 1 = 0 : 7 ? +3@53. ' !" 0

(0, 0, . . . , 0) )

! ' # E ' ' (

' " )

0 ' a11 − λ a21 .. . an1

a12

+3@53. '

a1n

a22 − λ . . . .. .

a2n .. .

an2

. . . ann − λ

= 0,

+3@54.

0 λ

+3@54. ' #

+3@53. ' ) A = (aij )

x = (x1 , x2 , . . . , xn ) ∈ Rn :

Ax = λ x ,

:) ! A ' λ5 % x5

# K

% , !) -# 1(

λ1 , λ2 , · · · , λn ∈ R

A = (aij )

+3@54. 0

# Z λj % +3@53. ' ' ! . (j)

(j)

∈ S, x(j) = x1 , x2 , . . . , x(j) n

Ax(j) = λj x(j) ,

1 ≤ j ≤ n, 0(

! ϕ = 0 ' # 6 x(j)

) f

) ! # 6" !

) f

S ! ) ) f

(

' 0 V '#

$ +3@53. ' <%. '

0" !

x1 , x2 , . . . , xn 5 ' (

. 0 0) ' # 1

) n n aij xi xj − λ x2j = 0, i,j=1

7 (

j=1

) ' ? ) ! ' ' n f (x) ≡ f (x1 , x2 , . . . , xn ) = aij xi xj = λ i,j=1

կետերեոչեվ

ե վ եվե ցչ իա յո ,թ, λգվ րետեոեոցչ ի 353

կեչե եոնշվա եջրշվքվղ A չեդոշ շ ր, ե եվ եո ,ք իա եե Ax = λ x կետերեոցչշ վոեվ կեչեեդեր եվյպ x ր, ե եվ տ, դյո տ,ո վ,զյտ յո,ր f (x) :ցվ շեջշ եոնցչ,վդա րդեվեվք կ,վ λ եոԿ

,ք ղ

փերվետյոե,րա λj ր, ե եվ եո ,քշվ կեչեեԿ

f (x) = λ

ր, ե եվ տ, դյոշ կեչեո րդեվեվք ՛ f x(j) = λj , 1≤j≤n : ,դ,ո f :ցվ շեջշ րյզյո րդե շյվեո ,դ,ով ,վա եե

դեր եվյպ x

Բեվշ x(j)

(j)

՛ max f (x) = max f x(j) = max λj , 1≤j≤n 1≤j≤n x∈S (j) ՛ = min λj : min f (x) = min f x 1≤j≤n

x∈S

1≤j≤n

3 1 հԲե-ե ցրեջշվ լճշ չեքրշչցչշ ճ չշվշչցչշ չերշվՆ Rn դեոե(ցթջեվ S = {x ∈ Rn : |x| = 1} չշետյո ր:,ոեջշ տոե րեկչեվտե( եչեջե եվ f (x) =

x ∈ S,

aij xi xj ,

(aij = aji ∈ R)

i,j=1

քե-ե ցրեջշվ լճշ չ,(ենցջվ հ յքոենցջվՆ եո ,ք կետերեո ի A = (aij ) չեդոշ շ ր, ե եվ եո ,քվ,ոշ չ,(ենցջվշվ հ յքոենցջվշվՆ

" %

& !

[ ' ) $ 9 $

Q $ 9

+'9 . 4%

9 . ) $

4 % ) # +

% +

%. "

% 9 & = & ) $ 9

+'9 . 4% " % ) $ + . = N% , ) $ $ ) $ (!"

+'9 .

? % F &

% ) $ %

''(

% '

' ) $ 4 "

' '

4 % ) $ %

4 % = #

$ R$ f (M ) ) $

" D ⊂ Rn

f ∈ C (D) T ! 9 ) $ 9 D

4 %

4 1 K f ) $ $ % D *

M1 , M2 , . . . , Mk D

$ % +.

4 2 A f ) $ 4 % 9 $ f (M1 ), f (M2 ), . . . , f (Mk ) :

4 3 K f ) $ 9

∂D ! max f (M ),

M ∈∂D

9 % D

min f (M ) = "

M ∈∂D

( ∂D !%

= ( % 9 4

4 4 A , $ (k + 2) 4 %& f (M1 ), f (M2 ), . . . , f (Mk ), max f (M ), min f (M ) : M ∈∂D

M ∈∂D

< $$ 9 % f > 9

4% D> 9 max f (M )

'9 % f > '9

4% D> 9 min f (M )

M ∈D

* $ ( 6 C& * 32 R2 '

x = 0,

y =1−x

y = 0,

' % % 0" D5# : !

+3@55. , !-

def

z = f (x, y) = xy − y 2 + 3x + 4y −→ max.min., !'

D !

(x, y) ∈ D,

') % ( # y

@r B(0, 1) @ @ @ D @ A(1, 0) -x r @r @ O @

10 D

⎧ ⎧ ⎨f = y + 3 = 0 ⎨y = −3 x M0 (−10, −3) : ⎩f = x − 2y + 4 = 0 ⎩x = −10 y

;

M0 (−10, −3) ∈ D " !

D !

' f (M0 ) # 70

f 5 % D !

' )

D ! ∂D " # : " " OA, OB, AB

'0#

1- C (x, y) ∈ OA ! y = 0, 0 ≤ x ≤ 1 f f (x, 0) = 3x # U" OA '

fx (x, 0)

= 3

max f (x, y) = f (A) = f (1, 0) = 3,

(x,y)∈OA

min f (x, y) = f (O) = f (0, 0) = 0 :

(x,y)∈OA

2- C (x, y) ∈ OB ! x = 0, 0 ≤ y ≤ 1 f f (0, y) = 4y − y

# :

fy (0, y)

(0, 2) ∈ D !

# 7 ?

f (O) = f (0, 0) = 0,

f 5

= 4 − 2y = 0 f 5

D !

OB '

f (B) = f (0, 1) = 3 :

3- C (x, y) ∈ AB ! y = 1 − x, 0 ≤ x ≤ 1 ) '

f (x, 1 − x) = x(1 − x) − (1 − x)2 + 3x + 4(1 − x) = −2x2 + 2x + 3 : :

f (x, 1 − x) = −4x + 2 = 0,

x= , y= ,

1 1 , 2 2

= Mmax

" # 7 1 1 , = : fmax = f (M0 ) = f 2 2

C" 0) ' ) 1 1 max f (M ) = max f (M ) = f , = , M ∈D M ∈∂D 2 2

min f (M ) = min f (M ) = f (0, 0) = 0 :

M ∈D

A 6

M ∈∂D

! "

9 4 5 ? % C& * 33 R2 '

def D = (x, y) ∈ R2 : x2 + y 2 ≤ 25 ,

)

def

z = f (x, y) = x2 + y 2 − 12x + 16y −→ max.min., A"0

⎧ ⎨f = 2x − 12 = 0 x ⎩f = 2y + 16 = 0 y

(x, y) ∈ D :

⎧ ⎨x = 6

M0 (6, −8) :

⎩y = −8

M0 (6, −8) ∈ D

) !

70 f 5 % D

) D !

∂D = (x, y) ∈ R2 :

x2 + y 2 = 25

" , - # :? ! ⎧ ⎨z = f (x, y) = x2 + y 2 − 12x + 16y −→ max.min. ⎩ϕ(x, y) def = 25 − x2 − y 2 = 0 :

$" f 5 D

L = L(x, y, λ) = f (x, y) + λϕ(x, y) = x2 + y 2 − 12x + 16y + λ(25 − x2 − y 2 ), λ5 D 0" ! # +. D L(x, y)

1 ' " ) ') ' ⎧ ⎧ Lx (x, y, λ) = 2x − 12 − 2λ x = 0 x(1 − λ) = 6 ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ Ly (x, y, λ) = 2y + 16 − 2λ y = 0 y(1 − λ) = −8 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ Lλ (x, y, λ) = 25 − x2 − y 2 = 0 : x2 + y 2 = 25 : 7

)

M1 (3, −4),

λ = −1,

M2 (−3, 4),

λ=3:

0 !

J ! ') f

)

M1 M2

max f (x, y) = max f (x, y) = f (−3, 4) = 125,

(x,y)∈D

(x,y)∈∂D

min f (x, y) = min f (x, y) = f (3, −4) = −75,

(x,y)∈D

(x,y)∈∂D

C& * 34 R3 ?% %

def D = M (x, y, z) ∈ R3 :

x2 + y 2 + z 2 ≤ 4, z ≥ 0 ,

)

(x, y, z) ∈ D def

u = f (x, y, z) = x2 + y 2 + z 2 − 2x − 2y − 2z −→ max.min. :

A"0

⎧ f = 2x − 2 = 0 ⎪ ⎪ ⎨ x fy = 2y − 2 = 0 ⎪ ⎪ ⎩ fz = 2z − 2 = 0

⎧ ⎪ ⎪x = 1 ⎨ ⎪ ⎪ ⎩

M1 (1, 1, 1) ∈ D :

y=1 z=1

M1 (1, 1, 1) ∈ D # ; ?

M1 '

. ) 0

# W) D " 0

def S + = M (x, y, z) ∈ R3 : x2 + y 2 + z 2 = 4, z ≥ 0 def

D0 =

M (x, y, z) ∈ R3 :

x2 + y 2 ≤ 4, z = 0

% ∂D = S + ∪ D0 # $ <! !( . ! ' ⎧ ⎨u = f (x, y, z) = x2 + y 2 + z 2 − 2x − 2y − 2z −→ max.min. ⎩ϕ(x, y, z) def = x2 + y 2 + z 2 − 4 = 0 : : . ) )

' " f 5 D

L = L(x, y, z, λ) = f (x, y, z) + λϕ(x, y, z) = = x2 + y 2 + z 2 − 2x − 2y − 2z + λ(x2 + y 2 + z 2 − 4), λ5 D 0" ! #

կետրտո չվ

վտ

L ցտիա վյ ,եվա թտվ իրեր գր ն վ շվջ

քվ իվղքրտո դ ց րտո շրեդյվ շվքվիվ գ ⎧ ⎧ ⎪ ⎪ = 2x − + 2λ x = L x(1 + λ) = 1 ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨L = 2y − 2 + 2λ y = 0 ⎨y(1 + λ) = 1 y ⎪ ⎪ ⎪ ⎪ Lz = 2z − 2 + 2λ z = 0 z(1 + λ) = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x2 + y 2 + z 2 = 4 : ⎩ ϕ = x2 + y 2 + z 2 − 4 = 0

վ ղ թ x = y = z > 0 դ շվքվիվ տ ցտ քրի ց ցք √ 2 2 2 + −1: M2 = √ , √ , √ ∈S , ր պ λ= 3 3 3 նվ զյվ :վյքվտվիվտ ո,ե րքցք պվԿվ վ

:վյքվտտ ,եց ր ց

դ M2 շտվ վԿթ ո,ե րքցք իրե պտցյփգ :վ ղր ց իվ հյ ք շրեվղթերտո f

ո Բիվր

ցտիա վտ D0 - լվտ Կ վ ր պ x + y 2 ≤ 4

z = 0ր ճրՆվ( ցք ա շրեթ f

ցտիա վտ :վ ղ եր,ո գտ(ցտցք)

u(x, y) = f (x, y, 0) = x2 + y 2 − 2x − 2y : D0 - լվտ տր ,ցք f (x, y, 0) ցտ

⎧ ⎨u = 2x − 2 = 0 x ⎩u = 2y − 2 = 0 y

խր լվ:ր, D0 - լվտ րղ

ցտիա վտ ք վյտ քրի ,եվա թտվ

⇐⇒

իրե

M3 (1, 1, 0) ∈ D0 :

Կ վ ∂D0 = (x, y, z) ∈ R3 : x2 +y 2 = 4, z =

0 - լվտվ Կ վ (վ զյվ ,եվտցք րտո :վյքվտվիվտ ո,ե րքցք /տ(

իվ: քրի շվԿվ,վ ցքթԿ) ⎧ ⎨f (x, y) = f (x, y, 0) = x2 + y 2 − 2x − 2y −→ max.min. ⎩ϕ(x, y) def = x2 + y 2 − 4 :

հյ, /տ(

գ ց ր ց շվքվ ի ի տ իվղքրտո f բ չվ

վտ

ցտիա վտ)

L = L(x, y, λ) = f (x, y) + λϕ(x, y) = x2 + y 2 − 2x − 2y + λ(x2 + y 2 − 4),

λ5 D 0" ! # E D L

" ) ') ' ⎧ ⎧ Lx = 2x − 2 + 2λ x = 0 x(1 + λ) = 1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ y(1 + λ) = 1 Ly = 2y − 2 + 2λ y = 0 ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 2 x + y2 = 4 : ϕ=x +y −4=0

; x = y ' √ √ M4 = ( 2, 2, 0) ∈ D0 ,

√ √ M5 = (− 2, − 2, 0) ∈ D0 , D

M1,2,3,4,5

)

λ = √ − 1, λ = −√ − 1 :

') f

f (M1 ) = f (1, 1, 1) = −3, √ 2 2 2 √ √ √ , , = −4 3 − 1 , f (M2 ) = f 3 3 3 f (M3 ) = f (1, 1, 0) = −2, √ √ √ f (M4 ) = f ( 2, 2, 0) = −4 2 − 1 , √ √ √ f (M5 ) = f (− 2, − 2, 0) = 4 2 + 1 , √ √ √ max f (M ) = f (M5 ) = f (− 2, − 2, 0) = 4 2 + 1 ,

M ∈D

min f (M ) = f (M1 ) = f (1, 1, 1) = −3,

M ∈D

C& * 35 \? *? ,'

p1 = 3 p 2 = 2

,B '" - ' ! . 0 ! ' # : ' B

# :? ! ' . (

25 '

# C

! ' . '

1 25 '

15 ' MS

# 7

B !

15 25 '

4 15

) 15

25 !

" B ' ) ?) # q2

r6A(0, 10) c c A cAr C(2, 9) c Ac A c A c c A c A c D A c A c c A c A c c A A B(5, 0) r Ar q1 A O A

D . 0)

q1 ' 25 ! (

15 ! q2 '# : !

. # C

)

R(q1 , q2 ) = p1 q1 + p2 q2 = 3 q1 + 2 q2 : J ! 15

! ' 4 25 ! '

2 q1 + 4 q 2

25 '

15 ! ' 1

25 ! ' 3 q1 + q2 J ! ' % 2 q1 + 4 q2 ≤ 40,

3 q1 + q2 ≤ 15 :

< . ⎧ ⎨z = R(q1 , q2 ) = 3 q1 + 2 q2 −→ max. ⎩2 q + 4 q ≤ 40,

3 q1 + q2 ≤ 15,

q1 ≥ 0,

q2 ≥ 0 : def

% ?

, !- D = OACB =

# 6" ' ) D ? ( O(0, 0), D

A(0, 10),

C(2, 9),

B(5, 0) :

R5 ∂R ∂R = 3, = 2# ,R5 ' - ∂q1 ∂q2 '

D ? R

70 R " # :) R

)

R(q1 , q2 ) =

- !

0' ) R5 ) D ?

D ? (

⎧ ⎪ 3 q1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2 q2 , ⎪ ⎪ 30 − 3 q1 , ⎪ ⎪ ⎪ ⎪ ⎩2 q1 + 20,

6"

(q1 , q2 ) ∈ OB,

(q1 , q2 ) ∈ OA,

(q1 , q2 ) ∈ BC,

(q1 , q2 ) ∈ CA,

# 70 R

') R

D ?

O, A, B, C

R(O) = R(0, 0) = 0,

R(A) = R(0, 10) = 20, R(B) = R(5, 0) = 15, R(C) = R(2, 9) = 24, max R(q1 , q2 ) = R(C) = R(2, 9) = 24 :

(q1 ,q2 )∈D

: ! B ' ?) % 24 , 24 '" - ' ) B 2 ' 15 ! 9 ' 25 !

+& :+&+

) : ; , & 3:

4 6+*7

4%* 4 9: &,+, *&/ 9: &,+,&/ 1@

f (x, y) = (x − 1)2 + 2y 2 D( &

f (x, y) = (x − 1)2 − 2y 2 D( &

(0, 0) % ,

fmin = f (−2, −1) = −2

f (x, y) = x3 + y 3 − x2 − 2xy − y 2 D( &

fmin = f (0, 0) = 0

f (x, y) = x2 − 2xy + 2y 2 + 2x D( &

(1, 0) % ,

f (x, y) = x2 − xy − y 2 D( &

f (x, y) = x2 − xy + y 2 D( &

fmin = f (1, 0) = 0

fmax = f (0, 0) = 0,

fmin = f

4 4 , 3 3

=−

f (x, y) = x3 − 2y 3 − 3x + 6y D( &

fmax = f (−1, 1) = 6,

fmin = f (1, −1) = −6

(1, 1), (−1, −1) % , 8@

f (x, y) = 4x + 2y − x2 − y 2 D( &

fmax = f (2, 1) = 5

f (x, y) = x3 + y 3 − 15xy D( &

fmin = f (5, 5) = −125,

(0, 0) % ,

10@

f (x, y) = x2 + xy + y 2 − 3x − 6y D( &

11@

fmin = f (0, 3) = −9

f (x, y) = x2 + 4y 2 − 2xy + 4 D( &

fmin = f (0, 0) = 4

12@

x 1 + +y y x D( & fmin = f (1, 1) = 3

13@

f (x, y) = x2 + xy + y 2 − 2x − y

f (x, y) =

D( & 14@

f (x, y) = x3 y 2 (6 − x − y), D( &

15@

fmin = f (1, 0) = −1 x, y > 0

fmax = f (3, 2) = 108

f (x, y) = x4 + y 4 − 2x2 − 2y 2 + 4xy √ √ √ √ D( & fmin = f ( 2, − 2) = f (− 2, 2) = −8 (0, 0)> ,

16@

f (x, y) = x3 + 3xy 2 − 15x − 12y D( &

fmin = f (2, 1) = −28

fmax = f (−2, −1) = 28

(1, 2), (−1, −2) % ,

x2 y 2 − 17@ f (x, y) = xy 1 − D( &

18@

fmax = f (1, 1) = f (−1, −1) = √ −1 fmin = f (−1, 1) = f (1, −1) = √

f (x, y) = 1 − (x2 + y 2 )2/3 D( &

fmax = f (0, 0) = 1

19@

f (x, y) = (x2 + y 2 ) e−(x D( &

2 +y 2 )

e T 9

fmin = f (0, 0) = 0 ( fmax = f

" T = {(x, y) ∈ R2 : x2 + y 2 = 1} 20@

1+x−y f (x, y) = 1 + x2 + y 2 D( &

21@

22@

24@

8 x + + y, x, y > 0 x y D( & fmin = f (4, 2) = 6 f (x, y) = (x2 − 2y 2 ) ex−y fmax = f (−4, −2) =

(0, 0)> ,

f (x, y, z) = x2 + y 2 + z 2 − xy + x − 2z 2 1 D( & fmin = f − , − , 1 = − 3 3 y2 z2 2 + + , x, y, z > 0 4x y z fmin = f , 1, 1 = 4

f (x, y, z) = x + D( &

25@

√

f (x, y) =

D( & 23@

fmax = f (1, −1) =

f (x, y, z) = x2 + 2y 2 + z 2 − 2x + 4y − 6z + 1 D( &

fmin = f (1, −1, 3) = −11

26@

f (x, y, z) = 2x2 + y 2 + z 2 − 2xy + 4z − x 1 1 , , −2 = − D( & fmin = f 2 2

27@

f (x, y, z) = x3 + y 2 + 2z 2 + xy − 2xz + 3y − 1 =− D( & fmin = f 1, −2,

28@

29@

f (x, y, z) = xyz(1 − x − y − z) 1 1 1 D( & fmax = f , , = 4 4 4 f (x, y, z) = 2 D( &

30@

31@

fmin = f (−1, −1, 1) = −3

f (x, y, z) = xy + yz + zx

(0, 0, 0)> ,

f (x, y, z) = 2x2 + y 2 + 2z − xy − xz D( &

34@

f (x, y, z) = x2 + 4y 2 + 6z 2 − 2xy + 6yz − 6z

D( & 33@

f (x, y, z) = (x + y + 2z) e−(x +y +z ) √ , √ ,√ = D( & fmax = f e 2 3 2 3 3

=− fmin = f − √ , − √ , − √ e 2 3 2 3

D( & 32@

x2 y 2 + − 4x + 2z 2 y z 1 1 1 , , =− fmin = f 4 4 4

(2, 1, 7) % ,

f (x, y, z) = 3 ln x + 2 ln y + 5 ln z + ln(22 − x − y − z) D( &

fmax = f (6, 4, 10) = 3 ln 6 + 2 ln 4 + 5 ln 10 + ln 2

4 6+*7

* 9: &,+, *&/ 9: &,+,&/ 35@

f (x, y) = xy D( &

fmax

x+y =1 1 1 , = =f 2 2

36@

D( & 37@

38@

39@

40@

x2 + y 2 + z 2 = 9

x + y + z = 12

fmax = f (1, −2, 2) = 9 (x, y, z > 0)

fmin = f (2, 4, 6) = 2 · 42 · 63 = 6912

x+y+z =5

xy + yz + zx = 8

fmin = f (2, 2, 1) = f (2, 1, 2) = f (1, 2, 2) = 4 4 7 4 7 4 4 4 4 7 , , =f , , =f , , =4 fmax = f 3 3 3 3 3 3 3 3 3

= $$ %& √ L

43@

fmin = f (−1, 2, −2) = −9

f (x, y, z) = x y z D( &

42@

x y + =1 2 3 18 12 , = =f 13 13

f (x, y, z) = x y 2 z 3 D( &

41@

fmin

f (x, y, z) = x − 2y + 2z D( &

fmin = f (−1, −2) = −5

x2 + y 2 = 1 4 3 = 11 fmax = f − , − 5 5 4 3 , =1 fmin = f 5 5

f (x, y) = x2 + y 2 D( &

fmax = f (1, 2) = 5

f (x, y) = 6 − 4x − 3y D( &

x2 + y 2 = 5

f (x, y) = x + 2y

xyz ≤

x+y+z ,

x, y, z ≥ 0 :

2 Z f = xyz ) $ % , x + y + z = S

) $ f (K, L) = K 3/4 L1/4

, - " K>

L> (4

% ; 1 4 1

(4 4

& 20

24 000

K 9 D( &

fmax = f (450, 300) ≈ 406, 62

%& !& '

L&&+, $: 6+*7

,' $+ 5%:& $+

&<:&/ & ' &+ 44@

f (x, y) = 1 + x + 2y .

x ≥ 0, y ≥ 0, x + y ≤ 1 ; D( &

46@

min f = f (0, 0) = 1

min f = f (0, −1) = −1

max f = f (1, 0) = 2,

f (x, y) = x2 y

x2 + y 2 ≤ 1 2 1 ,√ D( & max f = f ± = √ 3 3 3 3 min f = f ± , −√ =− √ 3 3 f (x, y) = x2 − y 2 D( &

47@

max f = f (0, 1) = 3,

x ≥ 0, y ≤ 0, x − y ≤ 1 D( &

45@

x2 + y 2 ≤ 1

max f = f (±1, 0) = 1

f (x, y) = x2 + y 2 − xy + x + y

min f = f (0, ±1) = −1 x ≤ 0,

y ≤ 0,

D( &

max f = f (0, −3) = f (−3, 0) = 6 min f = f (−1, −1) = −1

x + y ≥ −3

48@

D( & 49@

0≤x≤

f (x, y) = sin x + sin y + sin(x + y)

π π 3 √3 max f = f , = 3 3

f (x, y) = x3 + y 3 − 3xy D( &

0 ≤ x ≤ 2,

π ,

0≤y≤

min f = f (0, 0) = 0 −1 ≤ y ≤ 2

max f = f (2, −1) = 13 min f = f (1, 1) = f (0, −1) = −1

3: 6+*7

,' $+ 5%:& $+

&<:&/ & ' *%&&%

* ' &+ 50@

f (x, y) = 3x + y − xy D( &

51@

54@

max f = f (1, 2) = 17,

max f = f (1, 0) = 5,

max f = f (3, 3) = 6,

f (x, y) = x2 + y 2 − 2x − 2y + 8 D( &

min f = f (3, 0) = −3

x = 0, x = 1, y = 0, y = 2 min f = f (1, 0) = −3 x = 0, x = 1, y = 0, y = 1

f (x, y) = x2 + 2xy − y 2 − 4x D( &

55@

max f = f (0, 0) = f (3, 3) = 0,

f (x, y) = 5x2 − 3xy + y 2 D( &

min f = f (0, 0) = f (4, 4) = 0

y = x, y = 0, x = 3

f (x, y) = x2 + 2xy − 4x + 8y D( &

53@

max f = f (2, 2) = 4,

f (x, y) = xy − x − 2y D( &

52@

y = x, y = 4, x = 0

max f = f (0, 0) = 8,

min f = f (0, 0) = 0 y = x + 1, x = 3, y = 0 min f = f (2, 0) = −4 x = 0, y = 0, x + y = 1 1 1 , = min f = f 2 2

56@

f (x, y) = 2x3 − xy 2 + y 2 D( &

57@

x = 0, x = 1, y = 0, y = 6

max f = f (0, 6) = 36,

f (x, y) = 3x+6y −x2 −xy −y 2 D( &

min f = f (0, 0) = 0 x = 0, x = 1, y = 0, y = 1

max f = f (1, 1) = 6,

min f = f (0, 0) = 0

1 2 x − xy y = 8, y = 2x2 D( & max f = f (−2, 8) = 18, min f = f (2, 8) = −14

59@ f (x, y) = 2x + 3y + 1 y = 0, y = 9 − x2 D( & max f = f (0, 3) = 28, min f = f (0, 0) = 1 58@

f (x, y) =

60@

f (x, y) = 4 − 2x2 − y 2 D( &

y = 0, y =

max f = f (0, 0) = 4,

√

1 − x2

min f = f (−1, 0) = f (1, 0) = 2

3 S [1] K@?@ Q( 9$ H? ! I A 1 1970 Г.М. Фихтенгольц, ”Основы математического анализа”, Части 1,2, изд. 10, 9-е, Лань, С-Петербург, Москва, 2015, 2008 [2] K@?@ Q( 9$ HR) $ 9 % $I A 1 DA 1949 Г.М. Фихтенгольц, ”Курс дифференциального и интегрального исчисления”, Часть 1, изд. 8-е, Физматлит, Москва, 2003 [3] Л.Д. Кудрявцев, ”Краткий курс математического анализа”, Том 2, изд. 3-е, Физматлит, Москва, 2005 [4] Л.Д. Кудрявцев, ”Курс математического анализа”, Том 2, Высшее Образование, Дрофа, Москва, 2004 [5] С.М. Никольский, ”Курс математического анализа”, изд. 6-е, Физматлит, Москва, 2001 [6] В.А. Ильин, Э.Г. Позняк, ”Основы математического анализа”, Часть 1, изд. 7-е, Физматлит, Москва, 2003 [7] В.А. Ильин, В.А. Садовничий, Б. Сендов, ”Математический анализ”, Часть 1, МГУ, Москва, 1985 [8] F@\@ ? H? !I ? 12 ] 9 2009 2012

[9] W.F. Trench, ”The Method of Lagrange multipliers”, Trinity Univ., San Antonio, Texas, 2012 [10] ?@A@ ? H/-9

C ''( ) $ I : 9 2002 [11] Б. Гелбаум, Дж. Олмстед, ”Контрпримеры в анализе”, Мир, Москва, [12] Б.П. Демидович, ”Сборник задач и упражнений по математическому анализу”, изд. 18-е, МГУ, Москва, 1997 [13] И.И. Ляшко и др., ”Математический анализ в примерах и задачах”, Часть 2, Вища школа, Киев, 1977 [14] K@ K9 @ K =@ V K@ ? ;@ * H? ! ( 9I ? 2 DA 2014

[15] В.Ф. Бутузов и др., ”Математический анализ в вопросах и задачах”, Физматлит, Москва, 2002 [16] И.А. Виноградова, С.Н. Олехник, В.А. Садовничий, ”Задачи и упражнения по математическому анализу”, МГУ, Москва, 1988

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