tài liệu ôn thi ĐH&CĐ

CHƯƠNG 1: CÁC BÀI TOÁN VỀ HÀM SỐ BÀI 1. PHƯƠNG PHÁP HÀM SỐ I. TÍNH ĐƠN ĐIỆU, CỰC TRỊ HÀM SỐ, GIÁ TRỊ LỚN NHẤT & NHỎ NHẤT CỦA HÀM SỐ 1.y=fxab⇔ ( )   x x a b∀ < ∈  ( ) ( )   f x f x< 2.y=fxab⇔ ( )   x x a b∀ < ∈  ( ) ( )   f x f x> 3.y=fxab⇔ƒ′x≥∀x∈abƒ′x= ∈ab 4.y=fxab⇔ƒ′x≤∀x∈abƒ′x= ∈ab 5.Cực trị hàm số: !"# ( ) k x x f x ′ = ⇔ $%& k x 6. '(#)*&+!,&-! • './y=ƒx)01#02ab3"# ( )    n x x a b∈  45 [ ] ( ) ( ) ( ) ( ) ( ) { }   67 67    8 n x a b f x f x f x f a f b ∈ = [ ] ( ) ( ) ( ) ( ) ( ) { }   6  6     n x a b f x f x f x f a f b ∈ = • 9y=fx2ab3: [ ] ( ) ( ) [ ] ( ) ( )   6 8 67 x a b x a b f x f a f x f b ∈ ∈ = = • 9y=fx2ab3: [ ] ( ) ( ) [ ] ( ) ( )   6 8 67 x a b x a b f x f b f x f a ∈ ∈ = = • !;& ( ) f x x= α + β #0< [ ] 8a b (#)*&(#,& (=>a; b  b  j j j x x x − ε + ε i i i x x x − ε + ε a x II. PHƯƠNG PHÁP HÀM SỐ BIỆN LUẬN PHƯƠNG TRÌNH, BẤT PHƯƠNG TRÌNH 1.9?-@AB#:ux=vx)!<!<- ( ) y u x=  +* ( ) y v x=  2.9?-&@AB#:ux≥vx)! @=<!ABC+*@=  ( ) y u x= DE@F#0 <+*@= ( ) y v x=  3.9?-&@AB#:ux≤vx)! @=<!ABC+*@= ( ) y u x= DE@F%A*<+*@= ( ) y v x=  4.9?-@AB#:ux=m)!<! <-AGy=m+* ( ) y u x=  5.HIJux≥m>∀x∈K⇔ ( ) K 6 x u x m ∈ ≥ 6.HIJux≤m>∀x∈K⇔ ( ) K 67 x u x m ∈ ≤ 7.HIJux≥m?x∈K⇔ ( ) K 67 x u x m ∈ ≥ 8.HIJux≤m?x∈K⇔ ( ) K 6 x u x m ∈ ≤ III. Các bài toán minh họa phương pháp hàm số Bài 1. L<! ( )   Mf x mx mx= + − a.J:m@AB#:ƒx=?x∈283 b.J:m&@AB#:ƒx≤?>∀x∈28N3 c.J:m&@AB#:ƒx≥?x∈ [ ] 8M− Giải: a.H$@AB#:ƒx=5 ( ) ( ) ( ) ( )     M M  M   M    f x mx mx m x x g x m x x x = + − = ⇔ + = ⇔ = = = + + −  Oƒx=?x∈283: [ ] ( ) [ ] ( ) 8 8 6 67 x x g x m g x ∈ ∈ ≤ ≤  M  P m⇔ ≤ ≤ b.J∀x∈28N3: ( )   M f x mx mx= + − ≤ ⇔ ( )   Mm x x+ ≤ ⇔ ( ) [ ]  M  8N  g x m x x x = ≥ ∀ ∈ +  [ ] ( ) 8N 6  x g x m ∈ ⇔ ≥  Q< ( ) ( )  M   g x x = + − .#028N30R⇔ [ ] ( ) ( ) 8N  6 N P x g x g m ∈ = = ≥ c.J+*x∈ [ ] 8M− : ( )   M f x mx mx= + − ≥ ⇔ ( )   Mm x x+ ≥  OS ( ) [ ]  M  8M  g x x x x = ∈ − + TU(V.WXR5 Y9 x = :&@AB#:#E!   Mm = ≥ 0+Z? Y9 ( ] 8Mx ∈ :HIJ ⇔ ( ) g x m≤ ? ( ] 8Mx ∈ ( ] ( ) 8Mx Min g x m ∈ ⇔ ≤  Q< ( ) ( )  M   g x x = + − . ( ] 8M 0R ( ] ( ) ( ) 8M  M [ x Min g x g m ∈ ⇔ = = ≤ Y9 [ ) 8x ∈ − :   x x+ < 0HIJ ( ) g x m⇔ ≥ ? [ ) 8x ∈ −  [ ) ( ) 8 Max g x m − ⇔ ≥ J ( ) ( ) ( ) [ ]   M    8  x g x x x x − + ′ = ≤ ∀ ∈ − +  Q< ( ) g x 0 [ ) ( ) ( ) 8  MMax g x g m − = − = − ≥ α β b x a v(x) u(x) a b x y = m Kết luận:ƒx≥?x∈ [ ] 8M− ( ] )  8 M 8 [ m  ⇔ ∈ −∞ − +∞   U  Bài 2. J:m&@AB#:5 M M  M x mx x − − + − < ?>∀x≥ Giải:HIJ ( ) M  M N    M   M  mx x x m x f x x x x x ⇔ < − + ∀ ≥ ⇔ < − + = ∀ ≥  J ( ) [  [   N   N  N     f x x x x x x x x −   ′ = + − ≥ − = >  ÷   R# ( ) f x W \LHJ ( ) ( ) ( )   M      M M x f x m x f x f m m ≥ ⇔ > ∀ ≥ ⇔ = = > ⇔ > Bài 3. J:m&@AB#: ( )  N     x x m m m + + − + − > > x∀ ∈¡ Giải: OS   x t = > : ( )  N     x x m m m + + − + − > > x∀ ∈¡  ( ) ( ) ( )    N      N  N  m t m t m t m t t t t⇔ + − + − > ∀ > ⇔ + + > + ∀ > ( )  N    N  t g t m t t t + ⇔ = < ∀ > + + J ( ) ( )    N   N  t t g t t t − − ′ = < + + 0 ( ) g t  #0 [ ) 8+∞ R#R⇔ ( ) ( )    t Max g t g m ≥ = = ≤ Bài 4. J:m@AB#:5 ( )  [ Nx x x m x x+ + = − + − ? Giải: O]V?  Nx≤ ≤ H$IJ ( )  [ N x x x f x m x x + + ⇔ = = − + −  Chú ý:9F ( ) f x ′ #7U%&:<(#&@C@%^=)_ Thủ thuật:OS ( ) ( ) M        g x x x x g x x x ′ = + + > ⇒ = + > +  ( ) ( )   [ N    [  N h x x x h x x x − ′ = − + − > ⇒ = − < − −  `R#5 ( ) g x > +!W8 ( ) h x a+!.R ( )   h x > +!W ⇒ ( ) ( ) ( ) g x f x h x = W`R# ( ) f x m= ? [ ] ( ) [ ] ( ) ( ) ( ) [ ] ( ) 8N 8N  87  8 N  [  8m f x f x f f    ⇔ ∈ = = −     Bài 5.J:m &@AB#:5 ( ) M M  M  x x m x x+ − ≤ − − ? Giải:O]V? x ≥ 9X.+HIJ+* ( ) M  x x+ − > ;Ab &@AB#: ( ) ( ) ( ) M M  M  f x x x x x m= + − + − ≤  OS ( ) ( ) ( ) M M  M 8 g x x x h x x x= + − = + − J ( ) ( ) ( )     M c  8 M      g x x x x h x x x x x   ′ ′ = + > ∀ ≥ = + − + >  ÷ −    Q< ( ) g x > +!W x∀ ≥ 8 ( ) h x > +!W0 ( ) ( ) ( ) f x g x h x= W x∀ ≥ 4&@AB#: ( ) f x m≤ ? ( ) ( )    M x f x f m ≥ ⇔ = = ≤ Bài 6. J:m ( ) ( )  N c x x x x m+ − ≤ − + ?> [ ] Ncx∀ ∈ − Cách 1. HIJ ( ) ( ) ( )   N cf x x x x x m⇔ = − + + + − ≤ > [ ] Ncx∀ ∈ − ( ) ( ) ( ) ( ) ( ) ( )           N c N c x f x x x x x x x x − +   ′ = − + + = − + = ⇔ =  ÷ + − + −   d;@.0R#67 [ ] ( ) ( ) Nc  cMax f x f m − = = ≤ Cách 2.OS ( ) ( ) ( ) ( ) N c N c [  x x t x x + + − = + − ≤ =  J    Nt x x= − + + 4&@AB#:#E! [ ] ( ) [ ]   N 8[ N 8 8[t t m t f t t t m t≤ − + + ∀ ∈ ⇔ = + − ≤ ∀ ∈ J5 ( )   f t t ′ = + > ⇒ ( ) f t W0 ( ) [ ] 8 8[f t m t≤ ∀ ∈ ⇔ [ ] ( ) ( ) 8[ 7 [ cf t f m= = ≤ Bài 7. J:m   M c P M x x x x m m + + − − + − ≤ − + > [ ] Mcx∀ ∈ − Giải: OS M c t x x= + + − > ⇒ ( ) ( ) ( )   M c e  M ct x x x x= + + − = + + − ⇒ ( ) ( ) ( ) ( )  e e  M c e M c Pt x x x x≤ = + + − ≤ + + + − = ( ) ( ) ( )    P M M c e 8 M8M   x x x x t t   ⇒ + − = + − = − ∈   TU ( ) ( ) ( ) ( )  M8M  e  8   8 M8M  7 M M   f t t t f t t t f t f       ′ = − + + = − < ∀ ∈ ⇒ = =   R ( )   M8M  7 M    f f t m m m m m     ⇔ = ≤ − + ⇔ − − ≥ ⇔ ≤ − ≥ Bài 8. (Đề TSĐH khối A, 2007) J:m@AB#: N  M    x m x x− + + = − ?" Giải: O45 x ≥ $@AB#: N   M    x x m x x − − ⇔ − + = + +  OS [ ) N N       x u x x − = = − ∈ + +  4 ( )  M g t t t m= − + =  J ( )  c   M g t t t ′ = − + = ⇔ = Q<R0=   M m⇔ − < ≤ Bài 9. (Đề TSĐH khối B, 2007):LC#D5f*g m > @AB#: ( )   P x x m x+ − = − )Z>?@X? Giải:O]V?5 x ≥  H$@AB#:5 ( ) ( ) ( )  c x x m x⇔ − + = − ( ) ( ) ( )    c x x m x⇔ − + = − ( ) ( ) ( ) M  M   c M  f 7 c Mx x x m x x x m⇔ − + − − = ⇔ = = + − =  R ( ) g x m⇔ = >?V<. ( ) 8+∞ J;+;R5 ( ) ( ) M N  g x x x x ′ = + > ∀ > Q< ( ) g x ! ( ) g x )01+! ( ) ( )  8 ) x g g x →+∞ = = +∞ 0 ( ) g x m= >?∈ ( ) 8+∞  f;R m∀ > @AB#: ( )   P x x m x+ − = − ?@X? Bài 10. (Đề TSĐH khối A, 2008)J:m@AB#:>?" @X?5 N N    c  cx x x x m+ + − + − = Giải:OS ( ) [ ] N N    c  c 8 8c f x x x x x x= + + − + − ∈ Yhh 7Y J5 ( ) ( ) ( ) ( ) M M N N       8c   c  c f x x x x x x     ′ = − + − ∈  ÷  ÷ −   −   OS ( ) ( ) ( ) ( ) ( ) M M N N     8 c  c  c , xu x v x x x x x = − = − ∈ − − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         c u x v x x u v u x v x x  > ∀ ∈  ⇒ = =   < ∀ ∈   ( ) ( )        c   f x x f x x f ′  > ∀ ∈  ′ ⇒ < ∀ ∈   ′ =  9:HHJIJ?@X? ⇔ N  c  c M  cm+ ≤ < + Bài 11. (Đề TSĐH khối D, 2007): J:m?@AB#:? M M M M   [   [  x y x y x y m x y  + + + =    + + + = −    Giải:OS   8u x v y x y = + = +  ( ) ( ) M M M     M Mx x x x u u x x x x + = + − × + = − +!        8    u x x x v y y x x x y y = + = + ≥ = = + ≥ = 4?#E! ( ) M M [ [ P M [  u v u v uv m u v u v m + =  + =   ⇔   = − + − + = −    ⇔ u v )!?-@AB#:; ( )  [ Pf t t t m= − + = ?? ( ) f t m⇔ = ?   t t ,i   8 t t≥ ≥  d;@H.0-! ( ) f t +* t ≥  −∞ h  [ Y ∞ ( ) f t ′ – –  + ( ) f t Y ∞   jN Y ∞ 9:.0?? j    N m⇔ ≤ ≤ ∨ ≥ Bài 12.(Đề 1I.2 Bộ đề TSĐH 1987-2001): J:x&@AB#: ( )    <  x x y y+ + + ≥ >+* y∀ ∈¡  Giải: OS  <  u y y   = + ∈ −    xcYhf(x) N  c  c + HIJ ( ) ( ) ( ) ( )         6  u g u x u x u g u   ∈ −     ⇔ = + + ≥ ∀ ∈ − ⇔ ≥   Q< ( ) y g u= )!<G+*  u   ∈ −   0 ( )   6  u g u   ∈ −   ≥  ( ) ( )                   g x x x x x x g    − ≥ − + ≥ ≥ +   ⇔ ⇔ ⇔    + + ≥ ≤ −  ≥      Bài 13.L<    M a b c a b c ≥   + + =  LC#D5    Na b c abc+ + + ≥ Giải:HOJ ( ) ( ) ( )      N M  Na b c bc abc a a a bc⇔ + + − + ≥ ⇔ + − + − ≥ ( ) ( )    c [ f u a u a a⇔ = − + − + ≥ #< ( ) ( )     M  N b c u bc a + ≤ = ≤ = −  9A ( ) y f u= )!<G+* ( )   8 M N u a   ∈ −     J ( ) ( ) ( ) ( ) ( ) ( )     M      c [  8 M      N N f a a a f a a a= − + = − + ≥ − = − + ≥  0R# ( ) 8f u ≥ ( )   8 M N u a   ∀ ∈ −      f;R    Na b c abc+ + + ≥ OGC7.R# a b c⇔ = = =  Bài 14. (IMO 25 – Tiệp Khắc 1984):  L<     a b c a b c ≥   + + =  LC#D5 j  j ab bc ca abc+ + − ≤  Giải: ( ) ( ) ( ) ( ) ( ) ( ) ( )        a b c a bc a a a bc a a a u f u+ + − = − + − = − + − =  O ( ) ( ) ( )   y f u a u a a= = − + − +* ( ) ( )      N a b c u bc − + ≤ = ≤ = )!< G+*(#=> ( ) ( ) ( )   j     N j a a f a a  + − = − ≤ = <     +! ( ) ( ) ( ) ( ) ( )   M  j j          N N j N M M j f a a a a a− = − + + = − + − ≤  Q< ( ) y f u= )!<G+* ( )   8  N u a   ∈ −     +! ( ) j  j f < 8 ( ) ( )  j   N j f a− ≤ 0 ( ) j j f u ≤ OGC7.R#  M a b c⇔ = = =  Bài 15. LC#D5 ( ) ( )  Na b c ab bc ca+ + − + + ≤ ∀ [ ]   a b c ∈  Giải: H$&GC+]!;&ab, c  ( ) ( ) ( ) [ ]   N   f a b c a b c bc a b c= − − + + − ≤ ∀ ∈ O ( ) y f a= )!<G+* [ ] a ∈ 0 ( ) ( ) ( ) { } 67  8 f a f f≤ J ( ) ( ) ( ) ( ) ( ) [ ]  N   N8  N N N   f b c f bc f a a b c= − − − ≤ = − ≤ ⇒ ≤ ∀ ∈ Bài 16. L6k5 ( ) ( ) ( ) ( ) [ ]         a b c d a b c d a b c d− − − − + + + + ≥ ∀ ∈ Giải: H%^&GC+]!;&a, b, c, d, 5 ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ]            f a b c d a b c d b c d a b c d = − − − − + − − − + + + ≥ ∀ ∈ O ( ) [ ]  y f a a= ∀ ∈ )!<G0 [ ] ( ) ( ) ( ) { }  6 6    a f a f f ∈ = J ( ) [ ]      f b c d b c d= + + + ≥ ∀ ∈ ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( )         f b c d b c d g b c d b c d c d = − − − + + + ⇔ = − − − + − − + + O ( ) [ ]  y g b b= ∀ ∈ )!<G0 [ ] ( ) ( ) ( ) { }  6    b g b Min g g ∈ = J ( ) ( ) ( ) ( )   8     g c d g c d c d cd = + + ≥ = − − + + = + ≥ ⇒ ( ) ( ) [ ]   f g b b= ≥ ∀ ∈ f;R ( ) f a ≥ R@ BÀI 2. TÍNH ĐƠN ĐIỆU CỦA HÀM SỐ A. TÓM TẮT LÝ THUYẾT. 1.y=fxab⇔ƒ′x≥∀x∈abƒ′x= ∈ab 2.y=fxab⇔ƒ′x≤∀x∈abƒ′x= ∈ab Chú ý: J#<AB#:@$ZV/%11. 2.<(!lR m,]V?ƒ′x=∈ab CÁC BÀI TẬP MẪU MINH HỌA Bài 1. J:m ( ) ( )  c [   M  mx m x m y x + + − − = + #02+∞ Giải: !#02+∞⇔ ( )    j    mx mx y x x + + ′ = ≤ ∀ ≥ +  ⇔ ( )    j   j mx mx m x x x + + ≤ ⇔ + ≤ − ∀ ≥ ⇔ ( )  j   u x m x x x − = ≥ ∀ ≥ +  ( )  6 x u x m ≥ ⇔ ≥ J5 ( ) ( )   j        x u x x x x + ′ = > ∀ ≥ +  ⇒ux#02+∞⇒ ( ) ( )  j 6  M x m u x u ≥ − ≤ = = Bài 2. J:m ( ) ( ) M    M N M y x m x m x − = + − + + − #0M Giải. !W#0M⇔ ( ) ( ) ( )    M  My x m x m x ′ = − + − + + ≥ ∀ ∈  Q< ( ) y x ′ )01x=+!x=M0⇔y′≥∀x∈2M3 ⇔ ( ) [ ]     M Mm x x x x+ ≥ + − ∀ ∈ ⇔ ( ) [ ]   M M   x x g x m x x + − = ≤ ∀ ∈ +  [ ] ( ) M 67 x g x m ∈ ⇔ ≤ J5 ( ) ( ) [ ]     P  M   x x g x x x + + ′ = > ∀ ∈ + ⇒gx#02M3⇒ [ ] ( ) ( ) M  67 M j x m g x g ∈ ≥ = = Bài 3. J:m ( ) ( ) M    M  M M m y x m x m x= − − + − + #0 [ ) +∞ Giải: !W [ ) +∞ ⇔ ( ) ( )    M   y mx m x m x ′ = − − + − ≥ ∀ ≥  ⇔ ( )     c m x x x   − + ≥ − + ∀ ≥   ⇔ ( ) ( )   c    x g x m x x − + = ≤ ∀ ≥ − +  J5 ( ) ( )     c M    M x x g x x x − + ′ = = − +   M c M c x x x x  = = − ⇔  = = +   8 ( ) )  x g x →∞ =  JnHHJ⇒ ( ) ( )   67  M x g x g m ≥ = = ≤  Bài 4. ( ) ( ) ( ) M    j j    My x mx m m x m m= − − − + + − −  [ ) +∞ Giải: !W#0 [ ) +∞ ( )   M   j j  y x mx m m x ′ ⇔ = − − − + ≥ ∀ ≥ J ( )  j M Mm m ′ = − +V ( )  M M j   N m   = − + >     0 y ′ = ?   x x< HIJgx≥B]?')!5 J ( ) y x ′ ≥ > x∀ ≥ ⇔ [ )  G+∞ ⊂  ( ) ( )     [  [   M  M  M [    c   M m x x y m m m S m m ′ ∆ >   − ≤ ≤   ′ ⇔ < ≤ ⇔ = − + + ≥ ⇔ ⇔ − ≤ ≤     < = <   Bài 5. J:m ( )    x m x m y x m + − + + = − #0 ( )  +∞ Giải: !#0 ( )  +∞ ⇔ ( )     N     x mx m m y x x m − + − − ′ = ≥ ∀ > − ⇔ ( ) ( )      N       g x x g x x mx m m x m x m   ≥ ∀ > = − + − − ≥ ∀ >   ⇔   ≤ − ≠     Cách 1:Phương pháp tam thức bậc 2 J5 ( )    m ′ ∆ = + ≥ R#gx=?   x x≤  HIJgx≥B]?')!5 Jgx≥>∀x∈+∞⇔ ( )  G+∞ ⊂  ( ) ( )           c   M   M   M      m m x x g m m m m S m ′ ≤ ≤ ∆ ≥      ⇔ ≤ ≤ ⇔ = − + ≥ ⇔ ⇔ ≤ − ≤ −      ≥ + = − ≤    Cách 2:Phương pháp hàm số J5g′x=Nx−m≥Nx−a∀xa⇒gx#02+∞ Q< ( ) ( ) ( )    c   M   6   M   M      x g m m m g x m m m m m ≥    = − + ≥ ≤ −  ≥     ⇔ ⇔ ⇔ ≤ − ⇔    ≥ +  ≤     ≤ ≤   Bài 6. J:m ( ) ( )  N [ <  M M y m x m x m m= − + − + − + . x∀ ∈¡ Giải:\0=!<( ( ) [ N   M y m x m x ′ ⇔ = − + − ≤ ∀ ∈ ¡ ( ) ( ) [ ] [ N  M  8g u m u m u⇔ = − + − ≤ ∀ ∈ − Q< ( ) [ ]  8y g u u= ∈ − )!< G0R ( ) ( )  c P  N  M     g m m g m  − = − ≤  ⇔ ⇔ ≤ ≤  = − + ≤   Bài 7. J:m!       M N e y mx x x x= + + + W+*g x ∈¡ Giải: \0=!<(   < <  <M   M y m x x x x ′ ⇔ = + + + ≥ ∀ ∈¡  x  x  x  x xoYLJ ⇔ ( ) ( )  M   < <  N< M<   M m x x x x x+ + − + − ≥ ∀ ∈ ¡ ( ) [ ] M  N    M  m u u g u u⇔ ≥ − − + = ∀ ∈ − +* [ ] < u x= ∈ − J ( ) ( )   N      8   g u u u u u u u ′ = − − = − + = ⇔ = − = d;@HHJR#R0=!<(⇔ [ ] ( ) ( )  [ 67  c x g u g m ∈ − = − = ≤  Bài 8. L<! ( ) ( ) ( ) M      M  M y m x m x m x m= + + − − + +  J:mV<.-!%!DN Giải. TU ( ) ( ) ( )      M  y m x m x m ′ = + + − − + = Q<  j M m m ′ ∆ = + + > 0 y ′ =  ?   x x< 4<.-!%!DN [ ]     8 8 8 Ny x x x x x ′ ⇔ ≤ ∀ ∈ − =   m⇔ + > +!   Nx x− = J   Nx x− = ⇔ ( ) ( ) ( ) ( ) ( )           N   N M  c N   m m x x x x x x m m − + = − = + − = + + + ( ) ( ) ( ) ( )   N    M  m m m m⇔ + = − + + +  j c M j   c m m m ± ⇔ − − = ⇔ = Vb@+*  m + > R# j c c m + = B. ỨNG DỤNG TÍNH ĐƠN ĐIỆU CỦA HÀM SỐ I. DẠNG 1: ỨNG DỤNG TRONG PT, BPT, HỆ PT, HỆ BPT Bài 1. '.@AB#:5 [ M  M N x x x+ − − + =  Giải. O]V?5  M x ≤ OS ( ) [ M  M N f x x x x= + − − + =  J5 ( ) N  M [ M    M f x x x x ′ = + + > − ⇒fx#0 (   M  −∞    6SV(f −=0@AB#:fx=?%R&x=− Bài 2. '.@AB#:5   [ M  Px x x+ = − + +  Giải. H&@AB#:⇔ ( )   M  P [f x x x x= − + + − + = Y9  M x ≤ :fxp⇒+Z? Y9  M x > : ( )      M  M P [ f x x x x x   ′ = + − > ∀ >  ÷ + +   ⇒fx#0 ( )   M +∞ !f =0>?x= Bài 3. '.&@AB#:5 M [ N  [ j j [ M j Px x x x+ + − + − + − < q Giải. O]V? [ j x ≥ OS ( ) M [ N  [ j j [ M jf x x x x x= + + − + − + − J5 ( ) ( ) ( )  M N [ M N [ j M     [ M j M [ j N j [ f x x x x x ′ = + + + > + × − × − × − ⇒fx#0 ) [  j  +∞   6!fM=P0q⇔fxpfM⇔xpM f;R?-&@AB#:i<)! [ M j x≤ < Bài 4. '.IJ5 M     [ N M   [ j j  M c x x x x x x x x x x+ + + = + + − + − + q Giải. q ( ) ( ) ( ) ( ) ( ) M     [ N M   [ j j  M c x x x x x x x f x x x x g x ⇔ = + + + − − − = − + − + = Jfx+!g′x=−cx  +x−jp∀x⇒gx 9?-fx=gx)!<!<- ( ) ( ) +!y f x y g x= =  Q<fxW8gx.+! ( ) ( )   Mf g= = 0q?%R&x= Bài 5. J:m67 ( )  <     < m x x x x x x + + ≤ + + + ∀ q Giải. OS ( )    <   <   t x x t x x x= + ≥ ⇒ = + = + ⇒   t≤ ≤ ⇒  t≤ ≤ Vq⇔ ( )     m t t t t   + ≤ + + ∀ ∈    ⇔ ( )      t t f t m t t + +   = ≥ ∀ ∈   + ⇔ ( )   6 t f t m   ∈   ≥ Q< ( ) ( )      t t f t t + ′ = > +  0ft       ⇒ ( ) ( )   M 6   t f t f   ∈   = = ⇒ M  m ≤ ⇒ M 67  m = [...]... Bng bin thi n Nhỡn bng bin thi n suy ra: f ( x ) f ( 0 ) = 0 (pcm) x0+f 0+ f 0 a , b, c > 0 Bi 2 Cho 2 CMR: T = 2 a 2 + 2 b 2 + 2 c 2 3 3 2 2 2 b +c c +a a +b a + b + c = 1 2 2 2 a b c a b c Ta cú: T = 1 a 2 + 1 b 2 + 1 c 2 = a ( 1 a 2 ) + b ( 1 b 2 ) + c ( 1 c 2 ) x +f +0 f Xột hm s f ( x ) = x ( 1 x 2 ) vi x > 0 1 2 Ta cú f ( x ) = 1 3x = 0 x = 3 > 0 2 Nhỡn bng bin thi n f (... bin thi n suy ra: Hm s y = f (x) ch cú cc tiu m khụng cú cc i CT+ g > 0 m = 1 thỡ f (x) cú 3 nghim phõn bit x1 < x 2 < x 3 c) Nu g ( 0) 0 Nhỡn bng bin thi n suy ra: Hm s y = f (x) cú cc i nờn khụng tho món yờu cu bi toỏn Kt lun: m 1 7 , 1 + 7 U { 1} 3 3 xx1x2x3+f 0+00+f+ CT C x2 Bi 15 Cho hm s y = f ( x ) = x 4 + ( m + 3) x 3 + 2 ( m + 1) CT+ Chng minh rng: m 1 hm s luụn cú cc i ng thi. .. 0, x1, x2 Xột 2 kh nng sau: a) Nu m < 1 thỡ x1 x 2 = m + 1 < 0 x1 < 0 < x 2 Bng bin thi n Nhỡn BBT suy ra x CĐ = 0 b) Nu m > 1 thỡ x1 x 2 > 0 3 ( m + 3) < 0 x1 < x 2 < 0 v x1 + x 2 = 4 CT C CT+ xx1x20+f 0+00+f+ Bng bin thi n Nhỡn BBT suy ra x CĐ = x 2 < 0 CT Kt lun: Vy m 1 hm s luụn cú x CĐ 0 C CT+ Bi 4 ( thi TSH khi B 2002) 4 ( m 2 9 ) x 2 + 10 cú 3 im cc tr Tỡm m hm s y = mx + Gii Yờu cu... y = m vi th y = g(x) Nhỡn bng bin thi n suy ra ng thng y = m ct y = g(x) ti ỳng 1 im f ( x ) = 0 cú ỳng 1 nghim Vy hm s y = f (x) khụng th ng thi cú cc i v cc tiu Bi 7 Chng minh rng: f ( x ) = x 4 + px 3 + q 0 x Ă 256q 27 p 4 3 p Gii Ta cú: f ( x ) = 4 x 3 + 3 px 2 = x 2 ( 4 x + 3 p ) = 0 x = v nghim kộp x = 0 4 Do f (x) cựng du vi (4x + 3p) nờn lp bng bin thi n ta cú: 3 p 256q 27 p 4 0... 1, 1); C(1, 0, 1); D(0, 0, 1); A(0, 1, 0); B(1, 1, 0); C(1, 0, 0) Mt khỏc do x + y + z = 3 nờn M ( x, y, z ) nm trờn mt phng (P): x + y + z = 3 2 2 Vy tp hp cỏc im M ( x, y, z ) tho món iu kin gi thit nm trờn thit din EIJKLN vi cỏc im E, I, J, K, L, N l trung im cỏc cnh hỡnh lp phng Gi O l hỡnh chiu ca O lờn EIJKLN thỡ O l tõm ca hỡnh lp phng v cng l tõm ca lc giỏc u EIJKLN Ta cú OM l hỡnh chiu ca OM... 1+ t 1+ t 2 2 2 2 2 2 2t + 1 t ữ = m 1 + 1 t 2 ữ f ( t ) = ( 2t + 1 t 2 ) = 2m (2) 2 1+ t 1+ t Ta cú: f ( t ) = 2 ( 2t + 1 t ) ( 2 2t ) = 0 t = 1; t = 1 2 Bng bin thi n t11(t) 0+ (t)4 Nhỡn bng bin thi n suy ra: 04 t [ 1,1] (2) cú nghim 2 thỡ tMin] f ( t ) 2m tMax] f ( t ) [ 1,1 [ 1,1 0 2m 4 0 m 2 Vy (1) cú nghim x 2 , 2 thỡ m [ 0; 2] x 2 3x 0 Bi 5 Tỡm... 256 Bi 8 ( thi d b H khi A nm 2004) Tỡm m hm s y = x 4 2m 2 x 2 + 1 cú 3 im cc tr l 3 nh ca mt tam giỏc vuụng cõn Gii Hm s cú 3 cc tr y = 4 x ( x 2 m 2 ) = 0 cú 3 nghim phõn bit m 0 , khi ú th cú 3 im cc tr l A ( 0,1) ; B ( m,1 m 4 ) , C ( m,1 m 4 ) Do y l hm uuu uuu r r chn nờn YCBT AB AC = 0 m = 1 Bi 9 Chng minh rng: f ( x ) = x 4 6 x 2 + 4 x + 6 luụn cú 3 cc tr ng thi gc to O l... f1 ( x C ) = f 1 Nu xC [xA, xB] m[ 3, 3] thỡ Minf(x) = ữ= 2 4 m [3, 3] Khi ú Minf(x) > 1 m 2 10m + 13 < 0 3< m 1 1 < m < 5 + 2 3 Bi 10 ( thi TSH 2005 khi A) 1 1 1 1 1 1 Cho x, y, z > 0 ; x + y + z = 4 Tỡm Min ca S = 2 x + y + z + x + 2 y + z + x + y + 2 z Gii: S dng bt ng thc Cụsi cho cỏc s a, b, c, d > 0 ta cú: ) ( 16 ( a + b + c + d... 1 ì b 2 + 2 ì c 2 + 2 = 3.6 2 b c a 1 3 6 2 ì a 2 ì 2 b 1 2 ữ 2 ì b ì 2 ữ c 1 2 ữ 2 ì c ì 2 ữ a 6 ữ = 3 8 = 3 2 Min S = 3 2 ữ Nguyờn nhõn: 1 1 1 3 = = = 1 a + b + c = 3 > mõu thun vi gi thit a b c 2 Min S = 3 2 a = b = c = Phõn tớch v tỡm tũi li gii : Do S l mt biu thc i xng vi a, b, c nờn d oỏn Min S t ti a = b = c = S im ri: a=b=c= 1 2 1 2 2 2 a = b = c = 4 1 = 1 = 1 =4 a... ln 3) + 5 x ( ln 5 ) > 0 x Ă (x) ng bin Mt khỏc (x) liờn tc v x 0x0 1+f 0+ f f ( 0 ) = ln 3 + ln 5 6 < 0 , f ( 1) = 3ln 3 + 5ln 5 6 > 0 (x ) Phng trỡnh (x) = 0 cú ỳng 1 nghim x0 0 Nhỡn bng bin thi n suy ra: Phng trỡnh f ( x ) = 3 x + 5 x 6 x 2 = 0 cú khụng quỏ 2 nghim M f ( 0 ) = f ( 1) = 0 nờn phng trỡnh (1) cú ỳng 2 nghim x = 0 v x = 1 Bi 3 Tỡm m BPT: m 2 x 2 + 9 < x + m cú nghim ỳng x . <    Kết luận : Jn+!R#6fxa⇔ 325m1 +<<  Bài 10. (Đề thi TSĐH 2005 khối A) L<   x y z > 8    N x y z + + = J:6-`      x. + +   + + + ≥ =  + + + + +      = + + ≥ + + ⇒ =  ÷  ÷ + + + + + +     Bài 11. (Đề thi TSĐH 2007 khối B) L<   x y z > J:6-`       y x z x y z yz zx xy . M e M M xyz x y z xyz xyz S xyz x y z x y z x y z + + + + + + ≤ = × ≤ × = + + + + Bài 14. (Đề thi TSĐH 2003 khối B) J:(#)*&,&-!  Ny x x=