Chuyên đề Cực trị trong đại số

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  1. Cực trị trong đại số
  2. ơ ≥ ≥ =( − )   =() − =−+ =− −  +≤   = = = − + − +=−( ) +≥ ⇒ ≤ − + = − + =− +− + = = ≥
  3. =− + −+ =−() + + + =− ()() + +++ −+   =−−+()  −++  −     =−−+()  −  + ≥     −  ≥ ∀ ()− − ≥   = + = = ⇔ ⇔ = = = = = = =− − − + ( ) =− + − + + + = ()()= −− + +≥ + − = ± = ± ≥ ≠ =
  4. ơ ∈ ( ) ⇔ ơ = ( ) ⇔ ≤ ≤ = + ⇒ ⇔ ơ = ∈ R + ⇔ ơ + = ∈ R ⇔ ơ − + = ∈ R ⇔ ∆ ≥ ⇔ − ≥ ⇔ ≤ ⇔ − ≤ ≤ − + = + ⇔ ơ = ∈ R + ⇔ ơ − + −= ∈ R • = ⇔ ⇔ ≠ • ≠ ⇔ ∆ ≥ ⇔ − − ≥
  5. ⇔ − + +≥ e ơ − + + − = − = ± ee ⇔ = =  −  =  +  ( − ) ≥  −  = = ≥  +  − ⇔ơ = + ơ − − − = • ⇔ ≠ • ≠ ⇔ ∆= + −≥ ⇔+ − ≥ ⇔ − −≤ ⇔− + ≤≤+ + ≥ ≤ ≤+ + = + + = + + − + = + + ++ −+ = ++ −+ = + +
  6. = + = − − ≤ ≤ − ϕ + = + = ∈[] + + + + = − − + + = = ≤ = + + ⇔ − + = ≥ e < < a ueu • ≥ + ≥ ⇔ • + + + ≥ ⇔ + = +
  7. > > e   ≤  +    ≤ => ≥ ơ + ≥ ⇔ ơ − + − ≤ ≤ − − ≤ ⇔ ⇔ uou uu u u u −
  8. −  −  − +  +  −   ≤ = = − = ⇔ = − ou − − u + = ouuo o uouoo uu + = + + + ≥ = ⇔ = u ou u oooo uo + −
  9. − + + − − ≥ + = + = − − ⇔ = ⇔ = − = ouo ơ + + + + + + ơ + + + + ≥ = = + + ơ + + ≥ + + + ≥ +   + +  + +  + ≥ + +  + + +  + + ≥ () + + − = ⇔ = = = − + −
  10. ơ + + ơ + + + − − − = + + ơ + + = + + = = + + ++ ++ ++ ơ =+++ + + + + = = + = − + ⇒ ≤ e ≤ ≤ ⇔ + + + + + ơ + + + ≥ = ơ
  11. ⇒ ơ (− )(− ) ()()()− − − ≤ ≤ + + + + ≥ ⇒ ≤ ≤ e + ≥ = ơ + ≥ + ≥ = = = = + + + ≥ + ≥ ( + ) ++≥ ( ++ ) = ++ + + ≤ + + = = ++ ++ ++ ++ =⇔ = = = ()++ ≤( ++ ) ()+ + ≤
  12. + + ≤ + + ≤ + + + + ≤ + + ≤ ≤ = ⇔ = = = =+++ ( ++ ) =+ ( ++ ) − − + + = = + = + −= + −≥− ( ) ⇒ ≥ − + + = = − ⇔  + + = =+≥ = + = ≤ ≥ − ≥ − ≥ − + +++≥ = ⇒ − +≥− =− ⇔ = ≥ ⇔ = • • = = − + − ≤ ≤
  13. ≤ − − = ⇔ + + + + +   + + + + + − ()() + + = + +  + +  − + + + + +  + + +    []()()() + + + + +  + +  − + +  + + +  ( + + ) − () + + e + + + ( + + ) − () + + = = u o + + ≥ () + + − ( + + ) + + + o u uo + + + + + + +  +   +     +  +  +  +  +  −  +   +   +   + +   + +   + +    +   +   −  +   +   +    () + +  + +  −  + + +    []()()() + + + + +  + +  − ≥ (++ ) − =  + + +  ơ
  14. α βγ + + α + = + + β + = + γ + = + + α = β = γ = = + + + + + + + + + + + + + + + + + + − + + + + + + = + ( + −+ ) ( − ) = = + + ơ + + + = + − = + + ()+ () + () + ơ − + − − + + = −+ + − − + ( +) −( +) +=−≤ − +≤⇔( − )( −≤⇔≤≤ ) =⇔ = =± =⇔ = =±
  15.      + + +++++       () () ()  +  +  +       ≥ ++ ++ +   + + +    ⇒ + +  ()() ++≥++  + + +  + + ⇒ + + ≥ + + + (+ )( +≥+ ) ()       = +  () −+≥ () () −+   −     −      ⇒ ≥() + =+ =+ ⇔ − =⇔ =⇔=−+ − () − ⇔++=⇔+ () =⇔= − =+ ⇔= − = = =  >   = = + + + + + ơơ ()+ + + + ≥ = ()()()+ + + + + e ++≥ =
  16.  = =  + + +  = ⇔ = =  =   ⇔ = ==⇔ == e α ≥ −α + α α ≥ > e e e e u
  17. eo α ≥ β > α α α β ≥ − + β β = β ⇔ β = ⇔ α α α ≥ − + β β β u + e ( ) ≥ − + ( ) () ≥ − + () ⇒ () ≥ − + + = α  α  α α α −β β ≥ −  +  β  β o β + β = α ≥ β > α + α
  18. α = α = β + β α + α e α  α  α α α −β β ≥ −  +  β  β α  α  α α α −β β ≥ −  +  β  β  α  α α α α −β β α −β β − () + + () +  β  β β + β α −β α −β = α −β  α  α α α − () + +  β  β α −β  α  α α α − () + +  β  β oue ơ + ≥   • ơ      ± ≤ + ()   • = ( ) =( + ) () ⇔ ()()±+± ≤ ++ +
  19. =  (∈ ) = ≠  = () ++ () +   ()()()± = ∈ ( ) − ( ) = ( () ) ⇒ = () + −⇒ = + = − ++=++ () + ≥ ⇔≥ ++ ⇔ + = = + + = +++ −+⊥∀∈ − ≥ • = =  = = =   π • > ∆ =  π  =  e π =+− =+− ⇒ = −+ ơ = + + = + ≥ ⇒ +++ −+≥⇔≥ = = = = − + + −+
  20. =−+ −+ − − = − + + + ≥ ⇔≥ −++ ⇒ = − + + + ⇔() −+ () −=⇔= + = − ++ + + −  = + = + ≤ − ≤ ∈[ ] + ≥ = += ⇔− − = π ⇔ =−⇔=± + π π = = ± + π + = ()  + + = () = − + ++ + = ⇔+ ++ = ⇔() +− =   + + −
  21.    = = = −     ≤ + +  + = −  ⇔ = ⇔ +=− +⇒+=  −   =  =   + = −    + = ⇔ ⇒=− −−  = −   + =   + = ⇒ = − = =+ + + + = −++ − +    = +  − +  −    () = −++ +( +++) −−+( ) =+−() +++ ()
  22.  =( + ) + = − ( − ) = ⇔ = + = + ⇔ =     = −  ++ −  +          −   − −      ≥    = =−+  + +          =  +  =       − −−   −  =     ⇔() +=⇔= −    = +  − +  −    ( u ) u = + u ∈[ +∞ ) [ +∞ ) u −
  23. + = ⇒ = ⇒    ()()=+−++  +++−  ++        () − −  −     e ≥ → → → → ⇒ ≥ ⇒ ≥
  24. ơ e ≥ a uơ ơ = − +− ơ u + + = + + = + − + u= = + ≥ u= u + u + e u +∞ u = u= ⇒−≤ u ≤ u= + uu − u = = u = − =
  25. = + − = + − ≥  + − Khi − ≥ ⇔   ≤ − ⇒ =    + − Khi −≤ ⇔−≤ ≤ Khi ≤ − hoặc ≥ + − ⇒ =  Khi − ≤ ≤ − + + = + − =− + +  ≥ ≤ − =   − ≤ ≤ = − =− =+( ++ )( ++ ) − − =+ + + + + ≤ − ≥ −  − + + − ≤ ≤−   ∈ − −    + + − ≤ ≤− =  − − − − ≤ ≤− = + + Khi − ≤ ≤− =− − − Khi − ≤ ≤− = − =− = − =−
  26. ơ = −+−− + − ≤ ≤ = −++ −+ u=− + + + = − ≤ ≤− =−+−− −= − ≤ ≤ = − + ≤ ≤ = − ≤ ≤ = − = −+−− + − ≤ ≤ = ⇔− ≤ ≤− =− ⇔ ≤ ≤ = −+−+ −+− = + − =+= +≥ =−++= ≡ ⇔ =⇔= = =
  27. + =⇔ + =  = =  ⇔  =  =  + = ơ » = u=− ++⇔=− u + + ⇔= + u − ơơ = u − = » u ≡ = u − u ≡ = u − = − u ⊥ ⊥ ∆ = ∆ ⇒= = =  u = = u −  ⇔  − = u −  u = 
  28. uuu u ⊥ ∉ ∈ ∈ ∈ ≥ = ⇔ ≡ ≤ ⇒ ≤  = o ⊥ ⊥ ∈ ∈ ⊥ ∈ =  =  = o = ≥ = ⇔ ≡ ⊥ ⊥ ∈ + = + = + = + ⇔ ⇔ ≤ ơ ≤ = ⇔ ≡
  29. + + ⊥ ⊥ ⇒  ⇒ ⊥ ⊥ ⇒  + ⇒= ⇒+= ⊥ ⊥ ⇒   ⇒ = ⇒ = ∈ ⊥ ≤ ++ = ≤ = ⇔ ≡ ⇔ ⊥ ∈ = = ⊥ ∈ ≤ = ≤ = ⇔ ≡ ⇔   = o   ⊥ ⇒ =  e + = = = + + + = + +
  30. ⇔ + ⇔ + ⇔  ⇔ ⇔ = + ⊥ ∈ =  =  = ⇒ = = = = ○ + = + = ≥ ⊥ ∈ + ≥ = ⇔ ≡ ⇔  ⇔ ⊥ ⇔  ○ ⇒  ơ     = o  ⊥ = = ⇔ ⇔  a ou • − < < + ≤ ⇔ ≤ • = = ≤ ⇔ ≤  • ≤ + = ⇔ ∈ ≥ − = ⇔
  31. ≤ + + ++ − = ⇔ − e ⇒ = = = = + ≥ + += + + = + + ≥ = = ⇔ ∈  e = = = + + = + + + + ≥ ≥ = ⇔ e  = ⇒ =
  32. ơ =   = = = = = + + + = + + + = +++ ≥ = ⇔ ⇔     ơ − ≤ ≤ + = = = ≤ ≤ ≤ = ⇔ ≡ ≡ ≥ = ⇔ ≡ ≡ − ≤ ≤ + − ≤ ≤ + = = = = = = −− ≤≤+ + ⇒ ≤ ≤ ≤ = ⇔ ≡ ≡
  33. ≡ ≡ ≥ = ⇔ ≡ ≡ ≡ ≡ ao ≥ ⇔ ≤ ⇔ ≤  ⇔ ≤ ⊥ = = ≤ ≤ = ⊥ ⊥ ⇒  ⇒  = o ⊥  + ⇒ = + = = ⊥ =  = = ⇒ =  = ⇒
  34. ⇒ ⇒ ≤ ≤ = = ⇔ ⇔ ⊥ ⇔  + + = ⇒   =  = o   ⇒ =  = o = −  =− o   =  −  =− o  =    =  =  =    = = + = + ⇒ + + = ⇒ ≤ + + ≤ = ⇔ ⇔ ơ + + ≠ = + + =  ⇒ =   ⇒  
  35. ⇒  ⇔+ ⇔+ ⇔ ⇔  = o +  = + = o  =  = = ⇔ = ⇔ ≠   = o −   = ⊥ ∈ ∈ ≤ = ⇔ ≡ ⇔ ⊥  ⇔  ⇔  ⇔ ⇔ ⇔ ≡ ⇔ ⊥ ơ ơ + ≥ ()+ ≤+( )( + ) +
  36. = − ≤ ≤ = = − + =+− = + − + = −+ +   = −  +≥   = ⇔ − = ⇔ =    = ( = o )  =  ơ ∼  ⇒=⇒ = = −= − = −− +   = − −  ≤   ≤ = ⇔ = ⇔ ⇔  ơ   = o e = + ơ = + = + = + ++= +++++ ++− + += ≥ =
  37. ơ + ≥ + ≥ + ≥ +++ ≥() +++ =  = = ⇔ ⇔ = = ( ) () +    =  ∼  ⇒=⇒ = =+()() −=−=− + = + − =− + ≥ = − = ⇔ ( ) ( ) +  = o  = o e + = = =≤ + + + =+ ≤+ ( + )
  38. = = = ≤ = ⇔ = ⇔ ⇔  = o
  39. Ư o ơ ơ ơ − ≤ ≤  − ≤o ≤ − ≤ ≤ ≥ ≥ kπ ⇔ ⇔ π ⇔ 3 kπ 3 ≤ ≤ ≥ ≥ ⇔ 2 ≥ − ≥ ⇔ 2 ≥ − ≥ π − 2 ⇔ − + 2kπ 2 π − ⇔ + 2kπ 2 π 2 − + 2kπ 2 π + 2kπ 2 ≤ ≤ ≤ ⇒ ≤ ≤ ⇒ ≤
  40. ≤ π   +    u π  π  π π   +   +   +        π  π  ≤  +  ≤  +      (a+ b )(1 − ab ) ∀ (1+a2 )(1 + b 2 ) uu + − + o + + + + − + + (a+ b )(1 − ab ) + ≤ + ≤ ⇒ ≤ ≤ (1+a2 )(1 + b 2 ) ơ
  41. []o + + o −   []− + o − −o − o −+ o −     − −  − − o −− − ≤     o − ⇔ − =  ⇔ =   ∆ a o ơ ∗ + + = ∗ + + = ∗ + + = ≥ + + ≥ ≤ + + ≤ ơ + + ≥ ⇔++ ≥ ++ ⇔++ ≥ ++⇔++ ≥ ⇔ + + ≥
  42. ()()++ ≥ ++ a     ++≥   + +      ⇔ + + ≥ + + ≥   ⇔ ≤   ⇔ ≤   ∆ +  −  + + + o o  o     +  −  +   +   o  o  ≤o  + o    =         +++ +−− = o o ≤ o ⇔ + + +≤ ⇔ + + ≤       eơ e e e e= e = e    (e+ e + e ) ≥       ⇔+o()()() e e + o e e + o e e ≥ ⇔ + + ≤⇒    e+ e + e ⇔ ∆ ⇔ ∆ ouo e + + ≥ + + ≤
  43. + + ≥ o + o + o ≥ + + ≤ o + o + o ≤ + + ≥ + + ≥ ∆ + + ≥ o ≤ ≤ ≤ + + ≤ + +
  44. − − − + + +   − +  +   + + =+ − ≥ − + + e + + ≥ () ()+ +  ()++( ++) ≥( ++ ) ≥     ⇔ + + ≥ () ⇒ + + ≥ ⇔
  45.  ≥  + = + u ơ +  +  + ≥   ≥       π  π   +   +  π  π  +  ≥     ≥               π π  + + ≥     π  ≥     ơ + + + + + + ≥ + ≥ + + + + + ≥ = + + + − () + o ≥ ⇒ ≥ + A A +
  46. ơ ≥ + + + + + + +     ≥ +  A      ≥ ơ + + + ≥ = + + + + + ∆ +     +  +  +         +  +  =+ + +      ≥ + +         = +  = +      o()() − − o +              ≥+ =+  ≥ +  −o +  + + ()         
  47.        ơ +  +  ≥+     +          +  +  +  +                   ≥ +   +   ≥ +  + +                     +  +  +  ≥ +  = +            +  ∆   uo () = o ()()u= u ()o = − ()() u = − u () = o ()u ()u = o u ==( ) + + + ==( ) ( − ) + ( −++ ) = +
  48. − ≤ ≤ =( ) = + = ( ) = + =⇔ += ⇔( + ) =  = ⇔   = −  − ( ) ( ) − ==() − ++ + + − ≤ ≤ =() = −++ ++ − + = () = + −+ ++ ()− ++++ () −+ = ()()−+ ++
  49. = ⇔−() ++++ () −+= ⇔+() −+=− () ++= − ≤ ≤ ()+( −+=− ) () ( ++ ) ⇔  ()()+ − ≥  + = ⇔  − ≤ ≤   = − ⇔ ⇔=− − ≤ ≤ − ( ) ( ) + + + + =∀ ∈ = + = + =( ++ ) ( +− ) =+ −= + − + + − + = − + + = − ≤ ≤
  50. ( − + ) = = −≤≤ − + + () − ⇒ = =⇔= ()− + + − − π = =± +kπ k ∈ »   ⇔ = ⇒ = −  +=   +≤ ≤+ ⇔ ≤ ≤ = = ơ =+ + +    = +  + +    
  51. == + + π  = +  +    π = + << =( + )( − ) + = + = + ( + ) =++ + α β δ α+ β + δ = =+ α ++ β ++ δ =() + + = + ∆ + = + − + = + =( + ) +
  52. ơ = + ( − ) π = < < ∆ =+ ++ ++ π + + = + + =+ ++ + + + ≤+ + ++ ≤ + = = + =          +++    ≤+()   +  ++                ⇒+  ++  ≥ +++     
  53.   ≥ +      ≥+  ≥+=()   = = π  ⇔ =± + kπ  = = + + =− + ++() = + () π  = + −    π  π −≤ −≤⇔− ≤  −≤   π  ⇔−() ≤+  −≤+  ()   = + = − ( ) ( ) π  = +  +    =( − ) + π   −    π  −≤ −  ≤⇔− ≤≤+   = + = − π >
  54. = + ≥ = ≥  =  π π   ⇔ =⇔=∈   =    =( + )( − ) = −+ − = + ( − ) = + =() +   = +    π α= α = <α < =() α + α =− () α −≤() −≤⇔−≤α () −≤ α ⇔−≤ ≤ = = − + ( − ) + = = + + () − − + = = + −+ −+   = − +=  −+      ⇔ = ⇒ = −  +=  
  55. + − = + − = +   ≤ −  +   ≤− + +   ≤−  −  +≤    − =  = = = ⇔   =  =   + =( − ) + ⇒=−+( ) + + =−+( ) ( + − ) + + =−( ) + () − ++≥ − =  = π ⇔  ⇔==  − =  = = + α ++ β ++ δ ≤++ + α ++ β ++ δ ≤ + α + β + δ α+ β + δ = + α ++ β ++ δ ≤ = =
  56. + α =+ β =+ δ ⇔+ α =+ β =+ δ ⇒α = β = δ = ⇔α = β = δ =± =() + + π   = − +             π  =  −  +   π  π − ≥−⇒  − ≥−   π  ⇔ −≥−    ≤ ≤⇒ ≥ π  = −+  ≥−    π   −  = − ⇔      = π ⇔ = = − = + +  −  =  +     =()() + +−   
  57. =++( + + + +− + − + − ) =() + + =() + + =() + −    ≤ ≤⇔≤+ ≤ ⇔≤+( ) ≤ ⇔≤+ ( ) ≤ ⇔−≤() () + −≤  () −   ⇔ ≤ ≤ π ≤  + () − = ⇔   ()− =
  58.   + = ⇔  − =  π  = =  = −  ⇔ ⇔  π =  =  ∆ π π π  + =  −    ⇒ + ≤< ⇔ + −< + − ≠ ∀ ∈ + = + − ⇔( + −=+ ) ⇔ +−( ) =+ ⇔ ⇔ +()() − ≥+ ⇔ − +≥+ + ⇔ + +≤ −− −+ ⇔ ≤ ≤ − − − + + =( + ) −   =−   =−   −  =−  =+  
  59. = =   ()⇔=  +  +   ⇔=+ + ⇔ + =− ⇔ + ≥() − ⇔ − +−≤ −+ ++ ⇔ ≤ ≤ + + = + + ≤ ⇔ ≤ ⇔ + ≤ ⇔ + ≤ ⇔ = ⇔( += ) + ≠ −  + ⇔ +=  +   ⇔− +=+ + ⇔+() + =− ⇔ ⇔+()() +≥ − ⇔ + +≥ − + ⇔ − −≤ − + ⇔ ≤ + − = =
  60. = + = + = + ≤+( )() + ⇒ ≤() + π  +=  −≤    ≤  =  π ⇔ π  ⇔ =  −  =    π = = ≤ ≤ ≤ ≤  ≤ ≤ ⇒  ≤ ≤ ⇒= + ≤ + = ≥ ( − ) π = −
  61. +( − ) ≥() − −  ⇔  ≥() −   ⇒ ≥() − π ( − ) = ≤ ⇔ =− ⇔ = π  ∈  =   = =+ ++ = + + +   ≤ +  () + ++   ⇒≤ () += ⇔ + = + ⇔+ =+ ⇔+ =+ − ⇔ =⇔ = = π
  62. ⇔ + =π − ⇒ += π − + ⇒ = − − ⇔ + =− − ⇔ + + = ∆ ⇒ > + + ≥ ≥ ⇔ ≥ ⇔ ≥ ⇔ ≥ ⇔ = = ∆ π π ⇔ + = − π  ⇒() +=  −    + ⇔ = = − ⇔() + =− ⇔ + + = ≤+++( )( ++ ++ ) =() + = ⇔ = = π π ⇔ = = = + − − += = ≤
  63. +  +  ≥   =   +  + ≤ ≤   e   + ⇔ ≤ + ơ ≤ + ≤ + + ≤ + + + + ⇔ = ≤ + + = =  π  − ⇔ = = = =  = ⇔
  64. ≥ + + = − + − + = + + + − + − + + + = + − − + − () = − + + + = − + + + + + +
  65. + + + + + + + ++ ++ ++ ++ = − + = + + ơ + + + + + ơơ +  +  +   +  +      + + = = + + +−+ () +− () ()()+ − ()()+ + ơ
  66. ≤ ≤  ≤ ≤ +−− ++ ++−+ u( − ) + u( + ) + + ∆ = + + =+ ++ ++  >  + =     = +  + +      + ≥ ơơ + ≥ − +   −+=  −  +≥  
  67. ≥ + ≥ ơ − + ≥ + ≥ = + + + + + + ( + ) +=+= ≥ − − + + + + +   ()()()+++++   + +  ≥ + + +  ≥ + +  ≥ ⇒  ≥   = + + ≥ =+++++≥()()()()()() + + + e ()()()− − +
  68.   + +       ơơ + ≤+  ++     +  ≤  =   ⇒ ≥   ⇒ +  ≥  