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2, M2 9. > 13.4. 7 9 22z = (x ; 1)4 + (y + 2)4:444< 5  (13.1):@z = 4(x ; 1)3 = 0 @z = 4(y + 2)3 = 0@x@y x = 1 y = ;2@ M0(1 ;2): 5   . M0(1 ;2) :2 z @2A = @x2 = 12(x ; 1) = 12 (1 ; 1)2 = 0@MM002 z @B = @x@y = 0@M02 z @2C = @y2 = 12(y + 2) = 12 (;2 + 2)2 = 0M0M0 AC ; B 2 = 0:2, 9 .J 4z = f (x y) ; f (x0 y0) = (x ; 1)4 + (y + 2)2::, 8M (x y) 2 _(M0(1 ;2)) 4z > 0: 2, M0 , , :zmin = z x=1 = (1 ; 1)4 + (;2 + 2)4 = 0:y=;2> 13.5. 7 9 z = (x ; 1)4 ; (y + 2)4:< D M0(1 ;2):445E  M0 , 13.4, 0:A = B = C = 0:4AC ; B 2 = 0 .5 , _(M0) 9 M , 4z > 0 , M (x ;2)4z = (x ; 1)4 , 4z < 0 , M (1 y)4z = ;(y + 2)4:9 9Q_(M0) : 8M 2 _(M0) 4z > 0 ( 4z < 0), -, 9 M0 . > ):1) 7 67z = 2x3 + xy2 + 5x2 + y2 + 102) > 67 z = 2xy ; 3x2 ; 2y 2 + 103) > 67 z = x3 + 3xy 2 ; 15x ; 12y 4) > 67 z = (x + 1)4 + (y ; 1)4 :51) (0 0)032) zmax = 10 *; (;1 2) (;1 ;2)x = 0 y = 03) zmin = ;28 * x = 2 y = 1 zmax = 28 * x = ;2 y = ;14) zmin = 0 * x = ;1 y = 1:||||{446 14 z = f (x y) M0(x0 y0)4z = f (x y) ; f (x0 y0){ ,  M0(x0 y0) M (x y),  F (x y) = 0(14:1)  .= , z = f (x y) ( ) M0(x0 y0) _(M0) 4z < 0 (4z > 0) M 2 _(M0)  (14.1).A  ! .% ", ' "  (14.1)  y ( x), ..

 9 y = y(x) ( x = x(y)). = 9 f (x y) F (x y) = 0 () 9 f (x y(x)) = '(x)( f (x(y) y) = (y)). 14.1. 5 9 z = x2 + y2 , x y  x + y ; 2 = 0:< :   y z :y = 2 ; x@ z = x2 + (2 ; x)2 = 2x2 ; 4x + 4:F () 9 z = f (x):4475  z 0 = 4x ; 4 = 4(x ; 1): z 0 , :x = 1:5  9 :z 00 = 4 > 0:2, x = 1 z = f (x) .7  :x = 1 y = 1:F, z = x2 + y2 M (1 1):zmin = z x = 12 + 12 = 2:yJ . :z = x2 + y2 a ,  . 14.1.

A x + y ; 2 = 0 , a.5 { : . M0 0xy , z = 2  . >=1=1;. 14.1 14.2. 5 9 z = x1 + y1 , x y  x + y = 2:448< 7  x = 2;y z :z = 2 ;1 y + y1 () z = (2 ;2 y)y :F, 9 z = f (y):5  :z 0 = 2 (2 ;;y1)2 y2 (2 ; 2y) = (y4(;y ;2)21)y2 :- :z 0 = 0 y = 19Qz0 9Qz y = 0 y = 2;2, y = 1 z = f (y) .A , :x = 1 y = 1:A (1,1):zmin = z x = 11 + 11 = 2: >y=1=1 1. ) > 67 z = xy 2 * #, x y  x + 2y = 1:) > 67 z = x2 + y 2 * #, x y  x + 2.y =1:434491.

(1,0) { # , zmin = 0: 1, zmax = :27 36 481442. { # , zmin =:25 25251 1 { # 3 3|||||% ", ' " D  9 u = u(x y z ) F (x y z ) = 0: 14.3. 5 9 u = xyz , x y z  x + y + z = 5:< 7  z = 5;x;y u:u = xy(5 ; x ; y):F 9 u = f (x y):5  :@u = 5y ; 2xy ; y2 = 0 @u = 5x ; x2 ; 2xy = 0@x@y8 2< y + 2xy ; 5y = 0: x2 + 2xy ; 5x = 0::  , x2 ; y2 ; 5(x ; y) = 0 , (x ; y)(x + y ; 5) = 0:45028<866 y 2= x 2< (x ; y)(x + y ; 5) = 066 :8 x + 2x ; 5x = 0,62: x + 2xy ; 5x = 064 < y = 5 ; x: x2 + 10x ; 2x2 ; 5x = 028<66 y = x, 6666 :8< 3yx=(x5;;5x=3) = 04: x(5 ; x) = 0:,7 :!55M1(0 0) M2 3 3 M3(0 5) M4(5 0):5  :@ 2u = ;2y @ 2u = 5 ; 2x ; 2y @ 2u = ;2x:@x2@x@y@y2C D = AC ; B 2 .4 M1(0 0):A = 0 B = 5 C = 0 D = 0 0 ; 52 = ;25 < 0c, 9 M1 .!554 M2 3 3 :10 ; 10 = ; 5 C = ; 10 A = ; 10B=5;33 33325 = 75 > 0 A < 0;D = 1009! 9 9c, M2 35 53 { u = f (x y):7  z = 5 ; 35 ; 35 = 53 !555M 3 3 3 , umax = 53 53 35 = 12527 :4514 M3(0 5) :A = ;10 B = 5 ; 0 ; 10 = ;5 C = 0 D = 0 ; 25 = ;25 < 0@9 M3 .4 M4(5 0) :A = 0 B = 5 ; 10 ; 0 = ;5 C = ;10 D = 0 ; 25 = ;25 < 0@ M4 9 .7, umax = 12527!555 M 3 3 3 .

4 9 . > ) > 67 u = x3 y 2z * #, x3x + 2y + z = 12 (x > 0 y > 0 z > 0):y z  (2 2 2) { , umax = 64:||||{ 14.4. 5 2x ; z = 0 , A(1 1 1) B (4 3 4) .< M (x y z ) A M B M :~ j = q(x ; 1)2 + (y ; 1)2 + (z ; 1)2jMAq~ j = (x ; 4)2 + (y ; 3)2 + (z ; 4)2:jMB2 u = (x ; 1)2 + (y ; 1)2 + (z ; 1)2 + (x ; 4)2 + (y ; 3)2 + (z ; 4)2452u = 2x2 + 2y2 + 2z 2 ; 10x ; 8y ; 10z + 44:= ,  u = f (x y z ) , x y z  2x ; z = 0::   z = 2x z ,u = 2x2 + 2y2 + 2(2x)2 ; 10x ; 8y ; 20x + 44u = 10x2 + 2y2 ; 30x ; 8y + 44:: !392u = 10 x ; 2 2 x + 4 ; 10 94 + 2(y2 ; 4y + 4) ; 2 4 + 44!23u = 10 x ; 2 + 2(y ; 2)2 + 272:!327C 2 2 2 : 2!3, 2 2 : 9!3z = 2x = 3 2 2 3 : > 14.5. 2 .

' , (. 14.2)?<;. 14.2 S { , S = b + b +22l cos l sin = (b + l cos )l sin :453  P = b + 2l:F : P = b + 2l , b l  S = (b + l cos )l sin S = const b > 0 l > 0 0 < < ::  S ; l cos b = l sin , S ; l cos + 2l:P = l sin5  :@P = S ; 1 ! ; cos + 2 = ;S + (2 ; cos )l2 sin = 0@l sin l2l2 sin @P = S ; cos ! + l sin = ;S cos + l2 sin3 = 0@ lsin2 l sin2 7 8< ;S + (2 ; cos )l2 sin = 0: ;S cos + l2 sin3 = 0:A ; cos , l2((2 ; cos ) (; cos ) sin + sin3 ) = 0@2sin = 0l2 sin (;2 cos + cos2 + sin2 ) = 0 , 4 cos = 1=2:J sin = 0 0 < < cos = 1=2 , = =3:7  , = 3 P . - ,  = =3  S , , l b.

>454 1. ) * +#$+ +#$, '(+ * *#(S $, *.2. ) * +#$ *###** + !" V , '( $,'*0$.1. @ +#$2. !.p2Spp2S 2S:||||{455 15 ( ) 9 f (x y) F (x y) = 0:(15:1) L 9  . 2 L(x y ) = f (x y) + F (x y)  )  , .. 8 0>>< Lx0 (x y ) = 0Ly (x y ) = 0>>: L0(x y ) = 0:. (x0 y0 0) { 9 , (x0 y0) { ,  9 (15.1).# " '* )S L  ,    y x  F (x y) = 0 9 . 2 , L 9. 9 9 (x0 y0)  ,  .

15.1. : x2 + y2 = 2 x + y + z ; 4 = 0 .< C  : 9 z = 4 ; x ; y , x y z  x2 + y2 ; 2 = 0:2 LL = 4 ; x ; y + (x2 + y2 ; 2)456 L0x = ;1 + 2x L0y = ;1 + 2y L0 = x2 + y2 ; 2:4 x y 8>>< ;1 + 2x = 0;21 + 22y = 0>>: x + y ; 2 = 0::  x = 21 y = 21 x y , 2 = 14 , ==;11==22:2 : x = y = 1 = 12 @ x = y = ;1 = ; 21 : A 9  (1 1) (;1 ;1) ( 9 ).7  , ,, z = 4 ; x ; y . D , 9 , ,  , { .: (1,1)z = (4 ; x ; y) x = 2=1 (;1 ;1)y=1z = (4 ; x ; y) x=;1 = 6:y=;12, z = 2 z = 6: >. 9 f (x y z ) F (x y z ) = 0457 L L(x y z ) = f (x y z ) + F (x y z ) 4 :L0x = 0 L0y = 0 L0z = 0 L0 = 0:: L ,  .

15.2. .    (?;< R { ,H { ,h { .= ( V = R2 H + 13 R2 h pS = Rl + 2RH = R R2 + h2 + 2RH:F : , R H h p 2 2S = R R + h + 2H !h2V = R H + 3 = const:4582 LL = Rp"R2 + h2 + 2H + V;R2!#hH+3 :5 :@L = R p h ; R! @hR2 + h2 3@L = R (2 ; R) @H2p3@L = 4 R2 + h2 + 2H + p R2 ; 2R H + h !5 @R3R2 + h2!@L = V ; R2 H + h :@35 9 L 8 p 22>Rh=(R+h);(=3)R= 0>>< R (2 ; R) = 0p 2pR2 + h2) ; 2R (H + h=3) = 022>R+h+2H+R=(>>: V ; R2 (H + h=3) = 0:7 .p57 R = 2 h = p4 :5h1 R , H = 2 h:5 ,   h, .7  , , L , .

=  p5R = 2 H = 12 h: >459 @*$ ' ! 7# 6  , ! !#$ #$ # < # +# .#$ ! #< !$ (h = 2r):||||{460 16 z = f (x y)  D, L (. 16.1). = M m  (  : ), D , .. L:. L { (. . 16.1),  F (x y) = 0  9 f (x y) F (x y) = 0:(16:1).

L ,  , (. 16.2),  9 .;; . 16.1. 16.2+   4 M m :1) f (x y) D@2) L, 9 (16.1)@3)  L ( )@4) M m .,   16.1. 5  z = (x ; 1)2 ; y2 x2 + y2 4:461< F D, x2 + y2 = 4 .5 :@z = 2(x ; 1) @z = ;2y:@x@y: 9 x = 1 y = 0:2 M (1 0) D.5 x2 + y2 = 4 9. 4 9 LL = (x ; 1)2 ; y2 + (x2 + y2 ; 4):5 @L = 2(x ; 1) + 2x@ @L = ;2y + 2y@ @L = x2 + y2 ; 4:@x@y@4 x y 8>>< x ; 1 + x = 0;2y + y2 = 0>>: x + y ; 4 = 0 :x = 2 y = 0 = ; 21 @ x = ;2 y = 0 = 12 @ppx = 21 y = 215 = 1@ x = 12 y = ; 215 = 1::  :z (1 0) = (1 ; 1)2 ; 0 = 0z (2 0) = (2 ; 1)2 ; 0 = 1z (;2 0) = (;2 ; 1)2 ; 0 = 90 p 1!2 0 p15 121151z @ 2 2 A = 2 ; 1 ; @ 2 A = ; 27 p15 1 1 !2 0 p15 12 701z @ 2 ; 2 A = 2 ; 1 ; @; 2 A = ; 2 :462:   m = ; 72 M = 9:7, z = ; 27 z = 9: > 16.2. 5  z = x2 + 4xy ; 2y2 ; 12x + 10 A(1 1) B (5 1) C (1 3):< F D , , x = 1, y = 1, y = ; 12 x + 72 (.

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