BÀI TẬP CHỌN LỌC VỀ PHƯƠNG PHÁP HÀM SỐ Bài 1.  m        − ! " # mxxx =++−− "" $ "$" % Rm ∈ &# Giải '( ( ) 2 3 2 3 1 2 2 1f x x x x= − − + + ) ( ) f x *+,-./. ! 1 1 2   −     # ( ) 0 2 2 3 2 2 3 2 3 3 4 3 3 4 1 2 1 1 2 1 x x x x f x x x x x x x x   + + = − − = − +  ÷ − + + − + +   # ! 1 1 2 x   ∀ ∈ −      2 3 2 4 3 3 4 3 4 0 0 3 1 2 1 x x x x x x + > − ⇒ + > ⇒ + > − + + # 123 ( ) 0 0 0f x x= ⇔ = # 4567.3 ( ) ( ) 0 88 88 1 0 1 2 0 1 CÑ 3 3 22 2 4 x f x f x − + − − − 9:+,6567.)3;<  ! 1 1 2   −     3 3 22 4 2 m − ⇔ − ≤ < ( 1m = # Bài 2. 3 ( ) "  * "* " *%" *& = %"&− + + + − ≤ * =! $   ∈ +   Giải '( 2 t x 2x 2= − + ⇔ " −">* " −"* 4%"&⇔ − ≤ ≤ ≤ ∈ + + 2 t 2 m (1 t 2),dox [0;1 3] t 1 ?5 2 t 2 g(t) t 1 − = + +@ ≤≤" 0%& 2 2 t 2t 2 0 (t 1) + + = > + #12A.B )"C 9)6 ⇔ 6 2 t 2 m t 1 − ≤ + ∈B )"C  ⇔  [ ] ∈ ≤ = = t 1;2 2 m maxg(t) g(2) 3 12 ≤ " $ # Bài 3. m:3  " " "% D& E = $ =x m x m x− + + + − + = # Giải " " "% D& E = $ =x m x m x− + + + − + = " " "% D& E = $x m x m x⇔ − + + + = − " " $ = " "% D& E = % $& x x m x m x − ≥   ⇔  − + + + = −   " $ " " E x x x m x ≥   ⇔  − + =  −  FG,)-244+@ " " % & " E x x f x x − + = − " " "% E & 0% & %" E& x x f x x − ⇒ = − ?3 4567.3 x H ∞ = EI" $E J ∞ K J = H H = J  L "DIE J ∞ ;  "D %L! & E m   ∈ ∪ +∞     # Bài 4. -@+,M,3>" L *J D "* Giải §Æt t = cos2x ( ) 1 t 1− ≤ ≤ th× sin 2 x = 1 t 2 − ( ) ( ) ( ) 3 3 3 3 1 1 f ' t 4t t 1 8t t 1 2 2   = + − = + −   ( ) ( ) ( ) 2 2 1 2t t 1 4t 2t t 1 t 1 2   = + − + − + −   = ( ) ( ) 2 1 3t 1 7t 4t 1 2 − − + B¶ng biÕn thiªn Qua b¶ng biÕn thiªn ta cã : miny = 1 27 vµ maxy = 3. Bài 5. N 3 3 cos sinx x m− = % & & O5P>H  6& % &Q ; 4 4 x π π   ∈ −     # Giải &?>H )R, ( ) ( ) cos sin 1 cos sin 1x x x x− + = − '(> cos sinx x − !SP 2t ≤ # ( ) 2 , 2 2 x k k l x l π π π π  = +  ∈  = +   ¢ 6&% & ( ) ( ) cos sin 1 cos sinx x x x m⇔ − + =  '(> cos sinx x − !SP 2t ≤ # ? ; 0; 2 4 4 x t π π     ∈ − ⇒ ∈       #T3 3 3 2t t m− = # 4U2,+7)Q ; 4 4 x π π   ∈ −     P+,VP 2 ;1 2 m   ∈ ÷  ÷    # "  WK%& W%& H I$ J = H $ 1 27 Bi 6. m3 $ $ " " " " $ $ " =% & $ " =%"& x y y x x x y y m + = + + = : Gii 'SP3 " " = = " " = x x y y y '(t>xJ tB=!"C!% &t $ $t " >y $ $y " # X,f%u&>u $ $u " 67.B=!"C.3 % & >yy>xJ %"& " " " =x x m + = '( " v x= vB=! C%"&v " J"v >m# X,g%v&>v " J"v =! =! % & ! % & " [ ] [ ] axg v g v= = 12P+,VP m"# Bi 7. N:PYZ zyx )) 5< $ """ =++ zyx #-@6[ zyx zxyzxyA ++ +++= E # Gii Đặt zyxt ++= " $ &%"$ " " =+++++= t zxyzxyzxyzxyt . Ta có $= """ =++++ zyxzxyzxy nên $$\$ " tt vì #=>t Khi đó # E " $ " t t A + = Xét hàm số #$$) " $E " &% " += t t t tf Ta có = EE &%0 " $ " > == t t t ttf vì #$t Suy ra &%tf đồng biến trên C$)$B . Do đó # $ D &$%&% = ftf Dấu đẳng thức xảy ra khi # $ ==== zyxt Vậy GTLN của A là $ D , đạt đợc khi # === zyx Bi 8. ::3 %H$& * J%"H&*J$H>=#% & Gii 'P*=#(> x != % & R,%]$&J%"H& " J$H>= 2 2 2 3 3 1 t t m t t + = + %"& FG,W%&> 2 2 2 3 3 1 t t t t + + %=& ^26567. % & %"&= 5 3 3 m # Bi 9. m ( ) D D " D " " =x x x x m+ + + = . =! # " Gii D D " " " x c x x+ = +, " D " " #c x x= $ 9 ( ) " $ " " " $x x m + + = # '( "t x= # [ ] [ ] =! " =! =! # " x x t _ ( ) [ ] " $ " $ ) =! f t t t m t= + + = 6567. `<. = =! " " $ m # Bi 10. Cho hai số thực x, y thoả mãn : $ $ "x x y y + = + Tìm giá trị lớn nhất và nhỏ nhất của biểu thức: A = x + y. Gii Ta có : ( ) $ $ " $ "x x y y x y x y + = + + = + + + Đặt: ( ) * $ "x y a+ = + + + = Ta đi tìm điều kiện của a đê hệ phơng trình sau có nghiệm: ( ) $ " x y a x y a + = + + + = (I) Ta có hệ (I) ( ) % & % "& $ $ " x y a x y a + + + = + + + + = Đặt ! " % =! =&u x v y u v= + = + Ta có hệ phơng trình: ( ) ( ) " " " " " $ $$ $ $ $ " \ a u v u v uv a u v a a a u v a u v uv a + = + = + + = + + = + = = ữ Suy ra : u và v là nghiệm của phơng trình: " " $ = $ " \ a a t t a + = ữ ữ (*) Hệ (I) có nghiệm khi và chỉ khi phơng trình (*) có hai nghiệm t 1 , t 2 không âm " " = L ED = \ $ " = = \ $ E " = \ "a = a a S a a P a a + + + + hay \ $ " \ $ E " A + + . Vậy: \ $ " \ $ E! " MaxA MinA + = + = . Bi 11. 6 ( ) " " a b = " $ = x x x m x m + + + Gii X6 ( ) " " a b = % & " $ = %"& x x x m x m + + + ( ) bx #X<P+,VPc [ ] = !bx M<%"&# D ( ) ( ) ( ) [ ] " " " $ " " $ " % !b " =& " x x x x x m m do x x x − + ⇔ − + ≥ + ⇔ ≥ ∈ ⇒ + > + Od [ ] " " $ % & ! !b " x x f x x x − + = ∈ + X< [ ] = = !b 3 % &x f x m⇔ ∃ ∈ ≥ ( ) ( ) ( ) ( ) " " " " " D " " L 0 " " x x x x f x x x + − + − = = + + ! ( ) " a 0 = D = " f x x x x − ± = ⇔ + − = ⇔ = 1 [ ] !bx ∈ .V2 a " x − + = 3 " "a a $ a % & ) %b& ) $ $ " " f f f   − + − + = = =  ÷  ÷   1W-./+,,.B !bC. "a * % & $ f x = 9 [ ] [ ] = = !b "a !b 3 % & * % & $ x x f x m f x m m ∈ ∃ ∈ ≥ ⇔ ≥ ⇔ ≥ # Bài 12. O5 ( ) " " " $ $ $ x y x x y x y x x  + + + = −    + + = +  Giải 'P " =!x x y≥ ≥ −  >$PYI)Z;e+@^^X)  ( ) " " " " $ $ $ $ y x y x y x x x y x − = − ⇔ + − + = + − + )P7f%"&  " $ $ x x x+ + = ⇔ = -,+W%*&>1-YI6.%=!J ∞ &)+,>LI X % !L&# Bài 13. N ( ) ( ) $ D " " x x m x x x x m+ − + − − − = # Giải ; ( ) ( ) $ D " " x x m x x x x m+ − + − − − = % & 'SP3 = x ≤ ≤ g7  [ ] =! x ∈  M<% & ]*hM<% &.% &eSP " x x x= − ⇒ = # " x = +,% &f3 $ = "# "# " " m m m m =  + − = ⇒  = ±  i1@>=!% &R,3 ( ) " D D = " x x x− − = ⇔ = ;# i1@>H !% &R, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) D " " D D D " " " " = = x x x x x x x x x x x x x x x x x x + − − − − − = − ⇔ + − − − + + − − − = ⇔ − − + − − = J1@ D D = " x x x− − = ⇔ = J1@ = " x x x− − = ⇔ = jf,)% &h# E i1@> % &R,3 ( ) ( ) ( ) ( ) " " D D D " " x x x x x x x x x x+ − − − = − − ⇔ − − = − − % &" =) " x x= = .jf,% &PY# 12P>=+,>H # Bài 14. ?% ]E*& $= >  J *J " * " J#####J $= * $= kl_>8  8J"8 8J$8 " 8J###J$ 8 $= 8 Giải FGP3 $= = " " $= $= $= $= $= $= % E & #E #%E & ### #%E &x C C x C x C x− = − + − + gZ"+7+@*f3 $= = " " " $ $= $= $ $= $= $= $= % E & #E #E ### #Ex x C x C x C x C x− = − + − + % & ^,+7% &f! $= "\ = " " " $= $= $= $= $= $= $= % E & E= % E & " #E $ #E ### $ #Ex x x C C x C x C x− − − = − + − + %"& Nd*>H +,%"&f $= "\ = " " $= $= $= $= $= $= b E=#b "% #E& $% #E & ### $ % #E &C C C C+ = + + + +  "\ = " $= b %b E=& " $ ### $ a a a a+ = + + + +  $= = " $= b #"b " $ ### $ a a a a= + + + + Đ_3 $= b #"bS = . Bài 15. "Z63  x x m x x 2 2 10 8 4 (2 1). 1+ + = + + %$& Giải g2*G3 " " "  = L D "%" & "% &+ + = + + +x x x x %$&⇔ " " " " " " " = + +     − + =  ÷  ÷ + +     x x m x x #'( " " + = + x t x 'SP3–2< t E≤ #  mQm3m= " " "+t t #^2656..⇒ " D E < ≤m hoặc –5 < D< −m Bài 16. O5+,6-23 mx m x mx x x x 2 2 3 2 1.( 2 2 ) 3 4 2+ + + = − + − %D& Giải %D&⇔ $ $ % & % & % &+ + + = − + −mx mx x x # FG,3W%&> $ +t t ),,c67.m#  % & % &+ = −f mx f x  ⇔ + = −mx x  O5+,6-2.P7n5e#  • − < <m x = " − −m  •m = –1Q+@ ∀ ≥x  •Njfo-+Y# Bài 17. N*p=)p=M< " " $x y xy x y xy+ = + + #M6[  " " " % " & $ " xy P x y xy + − = + + # Giải  " " $x y xy x y xy+ = + +  % & $ % &xy x y x y xy⇔ + = + + *p=!p=.*Jp= % & ( ) " D $ $ $% & D =x y x y x y x y x y ⇒ + = + + ≥ + ⇒ + − + − ≥ + b ( ) [ ] % & D = Dx y x y x y⇒  + +  + − ≥ ⇒ + ≥   q,  ;  >  %*  J  & "  J  "  H  xy  ^    % &  $ xy x y ⇔ = + + $ x y xy ⇔ − = + g.;>%*J& " J J $ x y+ #'(*J>% D& ≥  " $ % &P t f t t ⇒ = + + =  0% &f t >"H $ " " $ " $ = pD t t t − = > ∀ , % &f t -./.rP5 [ ) D!+∞ g. % &f t c67.rP5 [ ) D!+∞ >p a % & %D& D P f t f= ≥ = #XM;6U a D P*>>"# Bài 18. 63 $ " % & % & x m x m x+ ≥ + + − + # Giải  $ " % & % & x m x m x+ ≥ + + − +  " " % &% & % & % &x x x m x x x⇔ + − + ≥ − + + +  " " " " " " 0 ) 0 = $ "% & x x t t x x x x x x − − + = = ⇔ = − + + − + − + # 4567. * H $− +  +∞ K J= H   $ " $ $ +  = = 12 a $ =! $ t   +   ∈     #4<P+,VP% & a $ =! $ t   +   ∈     P+,VP a $ B=! C $ * % & t M f t m + ∈ ≥ +@W%&>] " WK%&> ]" 4567.  = "  $ " $ $ + WK%& J =H W%&  D `6567.3 a $ B=! C $ * % & D " t M f t khi x + ∈ = = 126+@ D m ≤ # Bài 19. Na)b)c-,:PYZM< a b c+ + = #N[U3 a " "a ab bc ca abc+ + − ≤ # Giải a  " % & % " & % & % " &ab bc ca abc a b c a bc a a a bc+ + − = + + − = − + − #'(>6 " " % & % & = D D b c a t bc + − ≤ = ≤ = #FGW%&>% H&J% ]"&. " % & =! D a   −     NW%=&>% ]& " % & a D D "a a a+ − ≤ = < +, " " % & a a %" & D "a D $ $ "a a f a a   −   = − + − ≤  ÷  ÷  ÷     +@d [ ] =! ∈ 12 a " "a ab bc ca abc+ + − ≤ #'s[*5P>6>> I$ Bài 20. N*)∈m+,*)p #M ( ) ( ) $ $ " " % &% & x y x y P x y + − + = − − # Giải '(>*J!p"#t/4'D*≤%*J& "  " D t xy ≤ $ " %$ "& t t xy t P xy t − − − = − + #9$H"p=+, " D t xy− ≥ − . " $ " " " %$ "& D " D t t t t t P t t t − − − ≥ = − − + FG, " " " D % & ! 0% & ! " % "& t t t f t f t t t − = = − − WK%&>=⇔>=+>D# "D J∞ WK%&H= J W%&J∞ J∞  L 9;> %"! &  % &f t +∞ >W%D&>LfP D " D " x y x xy y + = =   ⇔   = =   Bài 21. N ) )x y z  [ ] =!" +, $x y z+ + = #-@ " " " A x y z= + + Giải O5r3 [ ] $ $ !"x y z x y z z z z≤ ≤ ⇒ = + + ≤ ⇒ ≥ ⇒ ∈ ^3  ( ) " " " " " " % & )%i& $ " b \ x y x y A z z z z + ≤ + ⇒ ≤ − + = − + FG [ ] " $ % & " b \) !" 0% & D b) 0% & = " f z z z z f z z f z z= − + ∈ ⇒ = − = ⇔ = $ \ % & E! %"& E! " " f f f   = = =  ÷   ?7f%i& 12 * EA = P =! ! "x y z= = = # L