GIÁ TRỊ LỚN NHẤT, NHỎ NHẤT CỦA HÀM SỐ VÀ ỨNG DỤNG I. DẠNG 1: TÌM GIÁ TRỊ LỚN NHẤT, NHỎ NHẤT CỦA HÀM SỐ Bài 1. (Đề dự bị TSĐH 2003 khối B)  ( ) 3 6 2 y x 4 1 x= + −   [ ] 1;1− Cách 1. [ ] 2 u x 0;1= ∈  ( ) 3 3 3 2 y u 4 1 u 3u 12u 12u 4= + − = − + − + [ ] 2 1 2 2 y 9u 24u 12 0 u 0;1 ;u 2 1 3 ′ = − + − = ⇔ = ∈ = >   4 maxy 4;min y 9 = = Cách 2.  ( ) ( ) 6 6 6 6 6 2 2 x sin u y sin u 4cos u sin u cos u 3cos u sin u cos u 3 4 maxy 4= ⇒ = + = + + ≤ + + = ⇒ =  !"#$%&'() 6 6 2 3 6 6 2 3 8 8 8 8 4 sin u 3 sin u sin u 27 27 27 27 3 4 4 4 4 4 4cos u 3 4cos u cos u 27 27 27 27 3  + + ≥ × × =     + + ≥ × × =   ( ) 6 6 2 2 8 4 4 12 4 y sin u 4cos u sin u cos u y 9 3 3 9 9 = + + ≥ + = = ⇒ ≥ * 2 4 x min y 3 9 = ⇒ =  Bài 2. (Đề thi TSĐH 2003 khối B)  2 y x 4 x= + − Cách 1: +,- [ ] D 2;2= − . 2 2 x y 1 ; y 0 x 4 x 4 x ′ ′ = − = ⇔ = − − 2 2 x 0 x 2 x 4 x  ≥ ⇔ ⇔ =  = −  /+,  maxy 2 2 ; min y 2= = − Cách 2:  ( ) ( ) x 2sin u,u ; y 2 sin u cosu 2 2 sin u 2;2 2 2 2 4  π π π   = ∈ − ⇒ = + = + ∈ −       . maxy 2 2 ; min y 2= = − Bài 3.0/+, 122)3 2 x 3 y x 1 + = + 0' a b c 1+ + = '&45 2 2 2 a 1 b 1 c 1 10+ + + + + ≥ Giải. 065 D = ¡ .  ( ) ( ) 2 2 1 3x 1 1 y 0 x y 10 3 3 x 1 x 1 − ′ = = ⇔ = ⇒ = + + ( ) ( ) x x x x 2 2 2 x 3 / x x 3 / x x lim y lim lim lim x x 1 1 1 x x →∞ →∞ →∞ →∞ + + = = = + +  !78 x x lim y 1;lim y 1 →+∞ →−∞ = = − 9 x−::y′;<−<y − :: x9=> y′;<−<y − 99 x<9f′<−<+<f? 9   2 x 3 y 10 maxy 10 x 1 + = ≤ ⇒ = +  0@,A0 2 x 3 10. x 1+ ≤ + B%&28 ( ) 2 2 2 2 2 2 2 2 2 x a : a 3 10. a 1 x b: b 3 10. b 1 x c: c 3 10. c 1 a b c 9 10. a 1 b 1 c 1 10 a 1 b 1 c 1  = + ≤ +   = + ≤ +   = + ≤ +   ⇒ + + + ≤ + + + + + ⇒ ≤ + + + + + Cách 2. ,%CDE-8 ( ) ( ) ( ) OA a;1 ; AB b;1 ;BC c;1= = = uuur uur uur F ( ) OC OA AB BC a b c ; 3= + + = + + uuur uuur uur uur G OA AB BC OA AB BC OC+ + ≥ + + = uuur uur uur uuur uur uur uuur H)78 2 2 2 a 1 b 1 c 1 10+ + + + + ≥ Bài 4. ' 2 2 x y 1+ = I-I ( ) 2 2 2 xy y S 2xy 2x 1 + = + +  ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 xy y 2 xy y 2 xy y S 2xy 2x 1 3x 2xy y 2xy 2x x y + + + = = = + + + + + + +  7y=<!=< *y≠< x t y = ⇒ ( ) ( ) ( ) 2 2 2 2 2y t 1 2 t 1 S 3t 2t 1 y 3t 2t 1 + + = = + + + +  5 ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 3t 2t 1 2 t 1 6t 2 2 3t 6t 1 S 3t 2t 1 3t 2t 1 + + − + + − + + ′ = = + + + +   )785 2 6 2 6 S 2 2 − + ≤ ≤  * 3 6 t 3 − − = ⇔ ( ) 3 4 6 3 6 y ; x y 20 3 − − − = ± = ×  2 6 MinS 2 − = * 3 6 t 3 − + = ⇔ ( ) 3 4 6 3 6 y ;x y 20 3 + − + = ± = ×  2 6 MaxS 2 + = Bài 5. (Đề 33 III.2, Bộ đề thiTSĐH1987 – 1995)' 2 2 x y 1+ = I-IA = x 1 y y 1 x+ + +  JA≤ ( ) ( ) ( ) ( ) 2 2 2 2 x y 1 y 1 x 2 x y 2 2 x y 2 2   + + + + = + + ≤ + + = +   * 1 x y 2 = =  A= 2 2 + J 7 xy 0≥ ⇒ [ ] [ ] x,y 0;1 A 0 x,y 1;0 A 1  ∈ ⇒ >  ∈ − ⇒ ≥ −   ⇒ Min A 1= − 1 x 1;y 0 x 0; y 1  = − =  = = −  : t−∞t 9 t : +∞!′−<+<−!< < 6K xy 0< 5 x y t+ = ⇒ 2 t 1 xy 0 2 − = < ⇒ ( ) t 1,1∈ − ( ) ( ) ( ) ( ) ( ) 2 2 2 A x 1 y 2xy 1 x 1 y y 1 x 1 xy x y 2xy 1 x y xy= + + + + + + = + + + + + + = 2 2 2 t 1 t 1 t 1 1 t 2 1 t 2 2 2 − − − + × + × + +  ( ) 2 t 1 1 2 t 2 1 2 −   = + + +   ⇔ ( ) ( ) ( ) 2 3 2 1 A f t 1 2 t 2 t 1 2 t 2 2 2   = = + + − + + −   5 ( ) ( ) 2 1 2 3 1 2 1 2 1 2 f t t 2 t 0 t t ;t t 2 1 2 2 3 + + + ′ = + − = ⇔ = = − = = − ( ) ( ) ( ) 1 2 2 19 3 2 f t ;f t 0 27 − = =  )785 ( ) ( ) 2 1 1 A f t A f t≤ ⇒ ≥ − ⇒ ( ) ( ) 1 2 19 3 2 Min A f t 1 27 − = − = − < −  -8   ⇔ 1 x y t+ = . 2 1 t 1 xy 2 − = ⇒xy2B 2 1 2 2 3 u u 0 3 9 + − + + = ⇒ ( ) 1 2 15 2 2 x,y 6 − + ± − = Bài 6.' [ ] x,y,z 0,1∈ L5 3 x y z 2 + + = M7&5 ( ) 2 2 2 S cos x y z= + +  Giải. G [ ] x,y,z 0,1∈  2 2 2 3 0 x y z x y z 2 2 π < + + < + + = < 122)3 y cos= α    ( ) 0, 2 π MI!I- ( ) 2 2 2 x y z+ + 6KBCN17(E-8O +,P,M ( ) M x,y,z LN7QB [ ] x,y,z 0,1∈ 4+,,RS 91T25 UV<990.WV9990.'V9<90.GV<<90.U′V<9<0.W′V99<0.'′V9<<0.EV<<<0 I  Q  #  3 x y z 2 + + =    ( ) M x,y,z  4     ,%VX05 3 x y z 2 + + = *+8+,P,M  ( ) M x,y,z  LN7QB  4 #B > t−9t 9 t : 9ƒ′+<−<+ƒ99 8 >=: E Y 9 9 F >=: Z I O - [ /  >=: 9 E′ Y[ZF/1MY[ZF/27M+,,RS \CE′2 7EY[ZF/5E′2] +,,RS12^2]$N7Y[ZF/ I2EI : _ 2 2 2 x y z+ + EI⇔E′I ⇔I`19aTY[ZF/H)785 ( ) ( ) ( ) 2 2 2 2 2 2 2 1 5 5 x y z OK 1 cos x y z cos 4 4 4 + + ≤ = + = ⇒ + + ≥   *+8  I  !  _   I ( ) ( ) 2 2 2 5 cos x y z cos 4 + + =  Bài 7. 'xb<2)3 ( ) ( ) ( ) ( ) ( ) 6 6 6 3 3 3 1 1 x x 2 x x f x 1 1 x x x x + − + − = + + + Giải. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 3 6 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 x x x x x x x x x x x x 1 1 f x x x x x 1 1 1 1 x x x x x x x x     + + + + − + + − +         = = = + − + + + + + + + ( ) ( ) 2 1 1 f x 3 x 3 x 6 6 x x   = + = − + ≥  ÷   *x_9IfVx0_a Bài 8. M7&PVxy0_x : ;99y : −axy;cx−:cy;:9 Giải. W d2d,RS PVxy0_Vx−>y;?0 : ;:Vy−90 : ;>≥> IXVxy0_>⇔ y 1 0 y 1 x 3y 4 0 x 1  − =  = ⇔   − + = = −    II. DẠNG 2: ỨNG DỤNG GTLN, GTNN ĐỂ GPT, GBPT Bài 1. \,RS5 4 4 x 2 4 x 2− + − =  ( ) 4 4 f x x 2 4 x= − + − 1 2 x 4≤ ≤ ⇒ ( ) ( ) ( ) 3 3 4 4 1 1 1 f x 4 x 2 4 x   ′ = −   − −   5  ( ) f x 0 x 2 4 x x 3 ′ = ⇔ − = − ⇔ =   )785 ( ) ( ) [ ] f x f 3 2 x 2,4≥ = ∀ ∈ ⇒ XRS    ( ) 4 4 f x x 2 4 x 2= − + − =    B #78x=> ? x−∞<x < 9+∞f′−<+f ƒVx < 0 Bài 2. \  ,RS  5  x x 3 5 6x 2+ = +  ⇔ ( ) x x f x 3 5 6x 2 0= + − − =     5 ( ) x x f x 3 ln 3 5 ln5 6 ′ = + − ⇒ ( ) ( ) ( ) 2 2 x x f x 3 ln3 5 ln 5 0 ′′ = + >  x∀ ∈¡ ⇒ƒ′Vx0e  IQƒ′Vx0$12 ( ) f 0 ln3 ln5 6 0 ′ = + − <  ( ) f 1 3ln 3 5ln 5 6 0 ′ = + − > ⇒XRSƒ′Vx0=<f9Bx < ⇒W   )785XRS  ( ) x x f x 3 5 6x 2 0= + − − =  Q(g7: B I2 ( ) ( ) f 0 f 1 0= = ,RSV90f:B x 0= 12 x 1= Bài 3. mMWX5 2 m 2x 9 x m+ < + Bf x∀ ∈¡ 2 m 2x 9 x m+ < + ⇔ ( ) 2 m 2x 9 1 x+ − < ⇔ ( ) 2 x m f x 2x 9 1 < = + − 5 ( ) ( ) 2 2 2 2 9 2x 9 f x 2x 9 2x 9 1 − + ′ = + + − =<⇔ 2 2x 9 9 x 6+ = ⇔ = ± ( ) x x 2 1 1 lim f x lim 9 1 2 2 x x →+∞ →+∞ = = + −  . ( ) x x 2 1 1 lim f x lim 9 1 2 2 x x →−∞ →−∞ − − = = + +  )785  ( ) ( ) 3 Min f x f 6 4 − = − =  M ( ) f x m>  x∀ ∈¡  ( ) x 3 Min f x m m 4 ∈ − > ⇔ < ¡ Bài 4. mMX5 ( ) 2 2 2sin 2x m 1 cosx+ = + V90B x , 2 2  π π ∈ −     G  : : x π π   ∈ −     ⇒ x , 2 4 4 −π π ∈      [ ] x tg t 1,1 2 = ∈ − ⇒ 2 2 1 t cosx 1 t − = + .  2 2t sin x 1 t = +   F    V90 ⇔ ( ) ( ) 2 2 2 sin x cosx m 1 cosx+ = + ⇔ ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2t 1 t 1 t 2 m 1 f t 2t 1 t 2m 1 t 1 t + − − = + ⇔ = + − = + +  V:0 5 ( ) ( ) ( ) 2 f t 2 2t 1 t 2 2t 0 t 1; t 1 2 ′ = + − − = ⇔ = = − ⇒W  h x−∞−aa+∞f′−<;<− ƒ t−99ƒ′Vt0−<+ƒVt0? <?  )785MV:0B [ ] t 1,1∈ −  [ ] ( ) [ ] ( ) t 1,1 t 1,1 Min f t 2m Maxf t ∈ − ∈ − ≤ ≤  ⇔ 0 2m 4 0 m 2≤ ≤ ⇔ ≤ ≤ *+8MV90B x , 2 2  π π ∈ −      [ ] m 0;2∈  Bài 5. m≥<MB5 3 2 2 35 sin xcosy m m 6m 4 33 cosxsin y m 6m 4  = − − +    = − +  V90B V90⇔ 3 3 2 sin xcosy cosxsin y m 12m 17 1 sin xcosy cosxsin y m 2m 2  + = − +   − = − +   ⇔ ( ) ( ) 3 3 2 sin x y m 12m 17 1 sin x y m 2m 2  + = − +   − = − +   V:0 6K  ( ) 3 f m m 12m 17= − +     5 ( ) 2 f m 3m 12 0 m 2 0 ′ = − = ⇔ = > !78ƒVm0≥ƒV:0=9∀m ≥< IQ# ( ) sin x y 1+ ≤ MBV:0Bm=: Q5 ( ) ( ) ( ) ( ) sin x y 1 2 3 1 sin x y 2  + =  ⇔  − =   8BV>0+ x ;y 3 6 π π = = 2B*+8V90BQm=: Bài 6. mMBWX5 2 3 2 x 3x 0 x 2x x 2 m 4m 0  − ≤  − − − + ≥  V90B V90⇔ ( ) 3 2 0 x 3 f x x 2x x 2 m 4m  ≤ ≤  = − − ≥ −  V:05 ( ) [ ) ( ] 2 2 3x 4x 4 x 0;2 f x 3x 4x 4 x 2;3  + − ∀ ∈  ′ =  − + ∀ ∈   .  ƒ′Vx0 = < ⇔ 2 x 3 =            )78  5 [ ] ( ) ( ) x 0;3 Maxf x f 3 21 ∈ = = MV:0B [ ] ( ) 2 x 0;3 Maxf x m 4m ∈ ≥ − ⇔ 2 m 4m 21− ≤  ⇔−>≤m≤i III. DẠNG 3: ỨNG DỤNG GTLN, GTNN CHỨNG MINH BẤT ĐẲNG THỨC Bài 1. '&45 ln x x< ∀xb< a x<:> f′−<++f<'c:9 m<: +∞ƒ′−<+ƒ9i9+∞ x<?+∞f′−<+f :−:: W⇔ ( ) f x x ln x 0= − > ∀xb<5 ( ) x 2 f x 0 x 4 2x − ′ = = ⇔ =  ⇒W  )785 ( ) ( ) f x f 4 2 2ln 2 0≥ = − > Bài 2. '&45 ( ) 2 2 1 xln x 1 x 1 x+ + + ≥ +  x∀ ∈¡ W⇔ ( ) ( ) 2 2 f x 1 xln x 1 x 1 x 0= + + + − + ≥  x∀ ∈¡   5  ( ) ( ) 2 f x ln x 1 x 0 x 0 ′ = + + = ⇔ =  ⇒ W      )785 ( ) ( ) f x f 0 0≥ = ⇒V,0 Bài 3. ' 2 2 2 a,b,c 0 a b c 1  >  + + =  'Ij5T= 2 2 2 2 2 2 3 3 a b c b c c a a b 2 + + ≥ + + + 5T= ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 a b c a b c 1 a 1 b 1 c a 1 a b 1 b c 1 c + + = + + − − − − − −  6K ( ) ( ) 2 f x x 1 x= − 1xb<)78 ( ) 2 1 f x 1 3x 0 x 0 3 ′ = − = ⇔ = >   ⇒ ( ) 2 f x x 0 3 3 ≤ ∀ >  F5 ( ) ( ) ( ) ( ) 2 2 2 2 2 2 3 3 3 3 a b c T a b c 2 2 f a f b f c = + + ≥ + + = Bài 4. '>≤nk'&45∀x≠<5 ( ) ( ) 2 n 2 3 n x x x x x 1 x 1 x 1 2! n! 2! 3! n! + + + + − + − + − <   ( ) ( ) 2 n 2 3 n x x x x x u x 1 x ; v x 1 x 2! n! 2! 3! n! = + + + + = − + − + −     A  &   ( ) ( ) ( ) f x u x .v x= l9 5 ( ) ( ) ( ) ( ) ( ) ( ) 2 n 1 n 2 n 1 n x x x u x 1 x u x 2! n! n 1 ! x x x v x 1 x v x 2! n! n 1 ! − −  ′ = + + + + = −  −  ′  = − + − + − = − − −  ⇒ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n x x f x u x .v x u x .v x u x v x u x v x n! n!     ′ ′ ′ = + = − − +         ⇒ ( ) ( ) ( ) [ ] ( ) n n 2 4 n 1 x 2x x x x f x u x v x 1 n! n! 2! 4! n 1 ! − − −   ′ = + = + + + +   −   G>≤nkƒ′Vx0`#7 1V−:x0⇒W ƒVx0 )785 ( ) ( ) f x f 0 1 x 0< = ∀ ≠ ⇒V,0 i x−∞<+∞f′−<+f < x−∞<+∞f′+<−f9 x−∞+∞f′+<−f BÀI TẬP VỀ NHÀ Bài 1. '∆UW' A B C> > 2)35 ( ) x sin A x sin B f x 1 x sinC x sinC − − = + − − − Bài 2. I-I5 y= 6 6 sin x cos x a sin xcosx+ +  Bài 3. 'ab≠<I ( ) 4 4 2 2 4 4 2 2 a b a b a b y b a b a b a = + − + + + Bài 4. ' 2 2 x y 0+ > I-I 2 2 2 2 x y S x xy 4y + = + + Bài 5. \)",RS 2 2 1 x px 0 p + + = Bx 9 x : p≠<) 4 4 1 2 S x x= +  Bài 6. I ( ) ( ) ( ) ( ) 2 x 2 x x x y 2 3 2 3 8 2 3 2 3   = + + − − + + −   Bài 7. 'xy≥<12 x y 1+ = I-I x y S 3 9= +  Bài 8. ' 2 2 2 x y z 1+ + = I-I P x y z xy yz zx= + + + + +  Bài 9. ' ( ) ( ) 3 2 f x cos 2x 2 sin x cosx 3sin2x m= + + − + I-IƒVx0H mM 2 [f(x)] 36 x≤ ∀ Bài 10 mMX5 ( ) ( ) 2 x 2 x 2 x 2 x m− + + − − + = B Bài 11 mMX5 2 x 9 x x 9x m+ − = − + + B Bài 12 mMX5 ( ) ( ) ( ) 6 5 4 3 2 x 3x m 6 x 2m 7 x m 6 x 3x 1 0+ − − − − − − + + = B Bài 13 mMX5 ( ) 3 2 2 2 x 2x 2 4 x 2x 2 2x 4x m− + − − + = − + ?B,] B Bài 14 mMX5 2 3x 1 2x 1 mx 2x 1 − = − + − B#78 Bài 15 mMX5 mcos2x 4sin xcosx m 2 0− + − = B ( ) x 0, 4 π ∈  Bài 16 mMX5 sin x.cos2x.sin 3x m= f:B x , 4 2 π π ∈      Bài 17 mMWX)7Bf x∀ ∈¡ 5 4 2 2 3cos x 5cos3x 36sin x 15cosx 36 24a 12a 0− − − + + − >  Bài 18 mMBWX5 2 2 3x 2x 1 0 x 3mx 1 0  + − <  + + <  B c Bài 19 a.mM5 2 m x 8 x 2+ = + :B,]B b.' a b c 12+ + = 'Ij5 2 2 2 a 8 b 8 c 8 6 6+ + + + + ≥ Bài 20 '&45 1 1 1 2 3 sin x sin2x sin 3x sin4x , x , 2 3 4 3 5 5 π π + + + ≥ ∀ ∈     Bài 21 '∆UW') 0 A B C 90< ≤ ≤ < ° '&45 2cos3C 4 cos2C 1 2 cosC − + ≥ Bài 22 '&45 3 2 sin 2x 3x x < −  ( ) x 0, 2 π ∀ ∈ Bài 23 '&5 ( ) ( ) 3 3 3 2 2 2 2 x y z x y y z z x 3+ + − + + ≤  [ ] x,y,z 0,1∀ ∈ Bài 24 '&45 ) : ) > ) ) ) : > x x nx x nx n + + +×××+ >  ( ) x 0, n π ∀ ∈ . 2 n≤ ∈¥ m . GIÁ TRỊ LỚN NHẤT, NHỎ NHẤT CỦA HÀM SỐ VÀ ỨNG DỤNG I. DẠNG 1: TÌM GIÁ TRỊ LỚN NHẤT, NHỎ NHẤT CỦA HÀM SỐ Bài 1. (Đề dự bị TSĐH 2003 khối B)  (. = MV:0B [ ] ( ) 2 x 0;3 Maxf x m 4m ∈ ≥ − ⇔ 2 m 4m 21− ≤  ⇔−>≤m≤i III. DẠNG 3: ỨNG DỤNG GTLN, GTNN CHỨNG MINH BẤT ĐẲNG THỨC Bài 1. '&45 ln x x< ∀xb< a x<:>. d2d,RS PVxy0_Vx−>y;?0 : ;:Vy−90 : ;>≥> IXVxy0_>⇔ y 1 0 y 1 x 3y 4 0 x 1  − =  = ⇔   − + = = −    II. DẠNG 2: ỨNG DỤNG GTLN, GTNN ĐỂ GPT, GBPT Bài 1. ,RS5 4 4 x 2 4 x 2− + − =  ( ) 4 4 f x x 2 4