Bài toán cực trị trong không gian hai chiều từ khía cạnh hình học

   »n  »  
 U f: U → . x0 ∈ U
aơa   
 f B( x0 , r ) x0
    a   ơ    
 r U fx()≤ fx ()0 fx()≥ fx ()0
 a aơa 
x∈ Bx(0 , r ). f( x 0 ) f.

  a  f U ⊂ »n 
  ơ  
x0 ∈ U . f x0 gradf ( x 0 )= 0
 ∂f
 (x )= 0;  i= 1, n 
 ∂x 0
  i
 
18  ƯỜĐẠỌỦĐÔỘ


 aa f 
aay
ya
a f a
y
 a
a  aơ
 f x= ( xx1 , 2 ,..., x n )
 (P ) a f ơ
 a
 gxx(1 , 2 ,..., x n )= 0
  f xx x
 min= (1 , 2 ,...,n )
  (P ) 
 gxx( , ,..., x )= 0
   1 2 n
    a         
 fxx(1 , 2 ,..., x n )
 a 
gxx(1 , 2 ,..., x n )= 0 (P )
  a
 x= ( xx1 , 2 ,..., x n )
 gradg ( x )≠ 0 (CQ ) 
  
  p( t )  g aa
  a  a 
 g( p ()) t = 0 t) x t0 p( t0 )= x ),
  ya 
 f( p ( t )) t= t 0. 0
 yya
 t= t 0,
 gradfpt (( )). pt′ ( )= grad fxpt (). ′ ( ) = 0 
  0 0 0
 y gradf ( x ) ya p( t ) 
x .yaa x 
gradf ( x ) a   x .ay
gradf ( x )  gradf ( x ). a λ 
a
 
 
ẠÍỌ −−−Ố   19 
 gradfx ( )= λ .grad gx ( )  (1) 
 
   Lx(,)λ= fx () − λ .() gx aaa (P ).  
 (1) ayaa
ư
  yaa  x  (P ) 
 (CQ )  λ 
  
 ∂L ∂ L ∂ L 
 grad(,)Lxλ= (,), x λλ (,),..., x (,) x λ  = 0 (2) 
  x x x 
  1 2 n  
 λ aax .
  yaaa
a x ya (P ) a
a g a(2) λ 
 yaaaơ
    a         
 z= fxx(1 , 2 ,..., x n )
 a
gxx(1 , 2 ,..., x n )= 0
 ư aa
 Lxx( , ,..., x ,)λ= fxx ( , ,..., x ) − λ gxx ( , ,..., x ) 
  12n 12 n 12 n
a (n + 1) 
 ư ơ
  ∂L
  (x , x ,..., x ,λ )= 0
 ∂x 1 2 n
  1
 ....................................
 
  ∂L 
  (x , x ,..., x ,λ )= 0
 ∂x 1 2 n
  n
 ∂L
  (x , x ,..., x ,λ )= 0
  1 2 n
  ∂λ
 
20  ƯỜĐẠỌỦĐÔỘ
  λ  a  ơ         
 (x1 , x 2 ,..., x n , )
 λ λ 
(x1 , x 2 ,..., x n , )
 ư  y    a   a  a  
 fxx(1 , 2 ,..., x n )
 aay
(x1 , x 2 ,..., x n )
 yaaơaaa
ay
   aơa fxyz(, , ) = x2 + y 2 + z 2   
ơ x2+2 y 2 − z 2 −= 10. 
 yaaa
 λ222 λ 2 22 
  Lxyz(,,,)(= xyz ++− )( x + 2 yz −− 1).
 a f aa
 Lxyz′(,,,)λ= 2 x − λ .2 x = 0 (1)
  x
 Lxyz′(,,,)λ= 2 y − λ .4 y = 0 (2)
  y
  λ λ 
 Lxyzz′(,,,)= 2 z −− (2) z = 0 (3)
 
  λ 2 2 2
 Lxyzλ′(,,,)= x + 2 y − z −= 1 0 (4)
  
   ơ aya
 (x0 , y 0 , z 0 ) z0 ≠ 0 (3)
λ = − 1.  ay    a λ    ơ  (1)   (2)  a  
 ơ  a
x= y = 0. (4). z0 = 0

   λ 
  (1;0;0) (− 1;0;0) = 1;
 1  1  1
 0; ;0  0;− ;0   λ = .
     2
  2  2 
 a fxyz(,,)= x2 + y 2 + z 2 . aaa
 1 
aaơ   aaa
 0, ,0 ,
  2  
 
 
ẠÍỌ −−−Ố   21 
 1  1
 ∆=ff0 + h , + klf ,0 +−  0, ,0  
   
  2  2
 2 2
 1   1
 =h2 ++ k +− l 2   
   
   2   2
 2 2 2
   =h + k + l + k 2. 
 1 
 ah k l aa
  ,+ , 
  2 
x2+2 y 2 − z 2 −= 10. a
 2
 1 
 h2+2 k + −−= l 2 1 0 ay h2+2 k 2 + 22 k −= l 2 0. 
  
  2 
 1 2
 aya k2= ( lhk −2 − 2 2 ). aya ∆f a
 2
 1
 ∆=+++fhklk2222 =+++ hkl 222 ( lh 222 −− 2) k 
  2
 3 1
 =l2 + h 2 ≥ 0. 
   2 2
 1 
        a
 0; ;0 
  2  
 1
min(,,)f x y z = . 
 2
 

 a
 2 2 
  uxy(,)= x + y ; (P )
 (x , y )  (∆ ) ơ
 
22  ƯỜĐẠỌỦĐÔỘ
  x+ y = 1. (CQ ) 
    aa Lxy( , ,λ )= x2 + y 2 + λ ( xy +− 1). 
aơ
 Lxy′(,,)λ= 2 x + λ = 0
  x
 Lxy′(,,)λ= 2 y + λ = 0 
  y
  λ
 Lxyλ′(,,)= x + y − 10 =
  
  
 1 1 
 a (,)x y =  ; . 
 2 2 
aaa
 2 2
 11  111  1 1
 ∆=uuhku +, +−  ,  =+++− h   k  
 22  222    2  2
    
 2 2
   =h + k + h + k 
  
 1 1 
   +h, + k a x+ y −1 = 0 
 2 2 
a
 1 1
 ++h +−= k1 0 ⇔ hk += 0. 
  2 2
  ∆=u h2 + k 2 ≥ 0. ya
a
 2 2
 1  1 1
 u(,) x y = +   = . 
 CT 2  2  2
   
          uxy( , ) = x2 + y 2    ơ
 M( x , y ) a »2 aO(0;0) 
y M( x , y )  x+ y = 1. 
y O(0;0) 
 
 
ẠÍỌ −−−Ố   23 
   x+ y = 1. ay 
a
 ax++ by c 1.01.01 +− 1
 d(,) O ∆=0 0 = = , 
 22 22 2
  a+ b 1 + 1
 
 2 1
 uCT (,) xy= dO (,) ∆ = 
  2
 ya (P ) ay (CQ ) a
a
  a
 2 2 
  uxy(,)= x + y ; (P )
 (x , y )  (∆ ) ơ
 2 2
  ax+ by = ca; + b ≠ 0. (CQ ) 
 aa
 λ2 2 λ
  Lxy(,,)= x ++ y ( axbyc +− ). 
 a
 Lxy′(,,)λ= 2 x + λ a = 0
  x
 Lxy′(,,)λ= 2 y + λ b = 0 
  y
  λ
 Lλ′(,,) x y= ax + by −= c 0
  
  
  ca cb 
 a (,)x y =  , . 
 a2+ ba 22 + b 2 
aaa
 ca cb  ca cb 
 ∆=uu + h, +− ku  ,  
 22 22  2222 
  ab+ ab +  abab ++ 
 2hca 2 kcb 2 c
 =++hk22 + =++ hk 22 ( hakb + ). 
 abab2222+ + ab 22 +
 
24  ƯỜĐẠỌỦĐÔỘ
  
  ca cb 
     +h, + k   a     
 ab22+ ab 22 + 
ax+ by − c = 0a
 ca  cb 
 a++ hb  +−= kc  0 
 22  22 
  ab+  ab + 
 ca2 cb 2
  ⇔ ++ah +−= bk c 0 
 ab22+ ab 22 +
  ⇔ah + bk = 0. 
  ∆=u h2 + k 2 ≥ 0 ya
a
 2 2
 ca  cb  c 2
 u(,) x y = +   = . 
 CT 22  22  22
  ab+  ab +  ab +
 ay (P )  O(0,0)  x+ y = 1
 H( m , n ) a
  a
 2 
  uxy(,)(= x − m )( + yn − ); (P )
 (x , y )  (∆ ) ơ
  x+ y = 1. (CQ ) 
 aa
 λ2 2 λ 
  Lxy(,,)(=− xm ) +− ( yn ) + ( xy +− 1).
 aơ
 Lxy′(,,)2(λ= xm − ) += λ 0
  x
 Lxy′(,,)2(λ= yn − ) + λ = 0 
  y
  λ
 Lxyλ′(,,)= xy +− 1 = 0
  
 
 
ẠÍỌ −−−Ố   25 
  
 mn−+1 nm − + 1 
 a(,)x y =  , . 
  2 2 
aaa
   
 mn−+1 nm −+ 1  mnnm −+−+ 11 
 ∆=uu + h, +− ku  ,  
 2 2   22 
 m+ n − 1
 =++hk2 2 2. ( hk + ) 
  2
  
 mn−+1 nm −+ 1 
   +h, + k a
  2 2 
x+ y = 1a
 mn−+1 nm −+ 1
 +h, + k = 1 
  2 2
 ⇔h + k = 0 
 
  ∆=u h2 + k 2 ≥ 0. ya
a
 2 2
 mn−+1  nm −+ 1  ( mn +− 1) 2
 uxy m  n  
 CT (,) = −+  −=  .
 2   2  2
 aaaa
  a
 2 2 
  uxy(,)(= xm − ) + ( yn − ) (P )
 (x , y )  (∆ ) ơ
 2 2
  ax+ by = ca; + b ≠ 0. (CQ ) 
 aa
 λ2 2 λ
  Lxy(,,)(=− xm )( +− yn ) + ( axbyc +− ) 
 aơ
 
26  ƯỜĐẠỌỦĐÔỘ
 Lxy′(,,)2(λ= xm − ) + λ a = 0
  x
 Lxy′(,,)λ= 2( yn − ) + λ b = 0 
  y
  λ
 Lxyλ′(,,)= ax +− by c = 0
  
 ơa
 2 2 
 mb−+ nab ca na − mab + cb 
 (,)x y =  , . 
 ab22 ab 22  
  + + 
  aaa

  2 2 
 mb−+ nab ca na −+ mab cb 
 ∆=uu + h, + k 
  ab22+ ab 22 + 
 
 2 2 
 mb−+ nab ca na − mab + cb 
 − u  , 
 ab22+ ab 22 + 
   
 2 2 2
   =++h k( c −− nb ma )( ha + kb ). 
 a2+ b 2
  2 2 
 mb−+ nab ca na −+ mab cb 
     +h, + k  a  
  ab22+ ab 22 + 
 ax+ by − c = 0a
 2  2 
 mb−+ nab ca  na −+ mab cb 
 a++ hb  +−= kc  0 
 ab22+  ab 22 + 
  2 
 222 2 2 
 mab−+ nab ca nab −+ mab cb 
 ⇔ ++ah +−= bk c 0
  22 22 
  ab+ ab + 
 c( a2+ b 2 )
 ⇔ ++−=ah bk c 0 
 a2+ b 2
 ⇔ah + bk = 0. 
 
 
ẠÍỌ −−−Ố   27 
 
 2 2
  ∆=u h + k ≥ 0. 
 yaa
 2 2
 mb2−+ nab ca  na 2 −+ mab cb 
 uxy m  n  
 CT ( , ) = −+  − 
 ab22  ab 22 
  +  + 
 ma22+ nb 22 ++ c 2 2 mnab − 2 mac − 2 nbc
 = 
 2 2
  a+ b
 2
 (ma+ nb − c )
 = .
 2 2
  a+ b
  y      uxy(,)(= xm − )2 + ( yn − ) 2    ơ
 M( x , y ) a »2  H( m , n ). 
y M( x , y )  ax+ by = c .
   y             
H( m , n )  ax+ by = c .ay
a
 ax++ by c am . +− bn . c
 d( H ,∆ ) =0 0 = 
 22 22
  ab+ ab +
 
 2
 (ma+ nb − c )
 u(,) xy= dH2 (,) ∆ = . 
 CT 2 2
  a+ b

 a  a  
aaơaa
 
28  ƯỜĐẠỌỦĐÔỘ
 

 ya  
 a
 yy  
 a

   a a        
  A
        a    ay