Chuyên đề khảo sát hàm số

Chuyên đề khảo sát hàm số, dành cho học sinh phổ thông thi đại hoc cao đẳng

WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam CHUYÊN ĐỀ 1: TÍNH ĐƠN ĐIỆU CỦA HÀM SỐ I.Kiến thức cơ bản: 1. Định lý:   xfDxxf ⇒∈∀>    xfDxxf ⇒∈∀<  2. Định lý mở rộng: Dxxf ∈∀≥      = xf  xf ⇒  Dxxf ∈∀≤      = xf  xf ⇒   3. Chú ý: ( ) baxxf   ∈∀> !"#$ [ ] ba xf ⇒  [ ] ba  ( ) baxxf   ∈∀< !"#$ [ ] ba xf ⇒  [ ] ba  4. Điều kiện không đổi dấu trên R: %&  ' ≠++= acbxaxxf     ≤∆ > ⇔∈∀≥    a Rxxf    ≤∆ < ⇔∈∀≤    a Rxxf    <∆ > ⇔∈∀>    a Rxxf    <∆ < ⇔∈∀<    a Rxxf II. Các dạng toán: 1. Tìm điều kiện để hàm số đơn điệu trên khoảng, đoạn cho trước: ()*+,+- ./0   %&0  1 %2,,3 Cách 1. 435)67830   9:;+< 9=3&<",>?7&, Cách 2.43@3)60  1>A"1 9BC=83" 9=3&<",>?7&, Cách 3.DE#)6)3, 9:;+<A)FAGH, 9=3&<",>?7&, Ví dụ 1.%& ( ) ( ) I ' J J ' J K I y x m x m x= − + + + + 3B,L B, ( ) ∞+ ' LTĐH - Chuyên Đề Khảo Sát Hàm Số JWWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam B, [ ] J I −  Giải: 3.;+",-1L  ( ) J'J' ' +++−= mxmxy ML    ≤∆ > ⇔∈∀≥⇔      a Rxy    J ' =⇔    = ∈ ⇔    ≤ > ⇔ m m Rm m .;+",-1L  ( ) J'J' ' +++−= mxmxy  ( )  ' J  ' J ' J  ' J x y x m x m x m =  = ⇔ − + + + = ⇔  = +  .)N6+J- ' J J m m+ = ⇔ =  .32<- " ∞− J ∞+   0  99  ∞+ 0 ∞−  I J  O03L ( ) ∞+ '  &21P3Q .)N6+'- JJ' >⇔>+ mm  .32<- " ∞− J'9J ∞+   0  9R9 0J ∞+ 0 ∞− 0'9J  =3&<3S0 ( ) ∞+ ' ' J 'J' >⇔>+⇔ mm P3?T .)N6+I- JJ' <⇔<+ mm  .32<- " ∞− '9JJ ∞+  LTĐH - Chuyên Đề Khảo Sát Hàm Số 'WWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam  0  9R9 0'9J ∞+ 0 ∞− 0J =3&<3S0?E2,&83  ( ) ∞+ ' U;0 ( ) ∞+ ' ?1&V ' J > m .;+",-1L  ( ) J'J' ' +++−= mxmxy  ( )    += = ⇔=+++−⇔= J' J J'J' ' mx x mxmxy .)N6+J- JJ' =⇔=+ mm  .32<- " ∞− J ∞+   0  99  ∞+ 0 ∞−  I J  O03L?E [ ] J I − &21?EP3Q .)N6+'- JJ' >⇔>+ mm  .32<- " ∞− J'9J ∞+   0  9R9 0J ∞+ 0 ∞− 0'9J  =3&<3S0?E2,&83  [ ] J I − .)N6+I- JJ' <⇔<+ mm  .32<- " ∞− '9JJ ∞+   0  9R9 0'9J ∞+ LTĐH - Chuyên Đề Khảo Sát Hàm Số IWWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam 0 ∞− 0J =3&<3S0 [ ] J I − 'IJ' −≤⇔−≤+⇔ mm .P3Q>?7W U;0 ' −≤ m  [ ] J I −  Ví dụ 2.%& J' I J 'I −−+= mxxxy 3B,L B, [ ) ∞+  B, ( ) J ∞− AB,&2AXJ Giải: 3.;+",-1L  mxxy −+= Y ' ML    ≤∆ > ⇔∈∀≥⇔      a Rxy Y YY J −≤⇔    −≤ ∈ ⇔    ≤+ > ⇔ m m Rm m .;+",-1L  mxxy −+= Y ' M [ ) ∞+   [ ) ∞+∈∀≥⇔   xy [ ) [ ) ∞+∈∀≥+⇔∞+∈∀≥−+⇔ Y Y '' xmxxxmxx BC xxxf Y ' +=  [ ) ∞+  .32 Y'  += xxf  '  −=⇔= xxf #& .32<- " ∞+ !  "9  ∞+  !"   =3&<3S0Z%[.  ≤⇔ m U;0  ≤ m  [ ) ∞+   LTĐH - Chuyên Đề Khảo Sát Hàm Số YWWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam .;+",-1L  mxxy −+= Y ' M ( ) J ∞−  ( ) J   ∞−∈∀≥⇔ xy ( ) ( ) J YJ Y '' ∞−∈∀≥+⇔∞−∈∀≥−+⇔ xmxxxmxx BC xxxf Y ' +=  ( ) J ∞− .32 Y'  += xxf  '  −=⇔= xxf ;  .32<- " ∞− R'J  !  "R9  ∞+  !"RY\  =3&<3S0Z%[. Y −≤⇔ m A.;+",-1L  mxxy −+= Y '   '  Y y x x m= ⇔ + − = M&2AXJ ⇔ +)*5  = ý 237+]7 'J ^ xx 3&& J 'J =− xx ( )    =−+ −> ⇔    =−+ >+ ⇔      =− >∆ ⇔ JY Y J' Y J  'J ' 'J'J ' ' ' J ' 'J  xxxx m xxxx m xx LTĐH - Chuyên Đề Khảo Sát Hàm Số \WWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam ( ) Y I Y I Y JY' Y ' −=⇔      −= −> ⇔    =−−− −> ⇔ m m m m m U;0 Y I −= m P3Q>?7&, Ví dụ 3.%& JJ' 'I −+−= xmxxy 3B,L B, ( ) +∞ J B, ( ) ' J AB,&2AX' Giải: 3.;+",-1L  J''I ' +−= mxxy ML    ≤∆ > ⇔∈∀≥⇔      a Rxy KK KK IK I ' ≤≤−⇔    ≤≤− ∈ ⇔    ≤− > ⇔ m m Rm m .;+",-1L  J''I ' +−= mxxy M ( ) +∞ J  ( ) +∞∈∀≥⇔ J  xy ( ) ( ) +∞∈∀ + ≤⇔+∞∈∀≥+−⇔ J J'I ' JJ''I ' ' x x x mxmxx BC ( ) +∞ + = J J'I  ' trên x x xf .32 ' '  J'I  x x xf − =    −= = ⇔= − ⇔= ' '  J'I  ' '  lx nx x x xf .32<- "J' ∞+ !  "R9 LTĐH - Chuyên Đề Khảo Sát Hàm Số KWWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam J\ ∞+  !" J'  =3&<3S0Z%[. KJ'' ≤⇔≤⇔ mm   U;0 K ≤ m P3Q>?7&, .;+",-1L  J''I ' +−= mxxy M ( ) ' J  ( ) ' J  ∈∀≤⇔ xy ( ) ( ) ' J J'I '' JJ''I ' ' ∈∀ + ≤⇔∈∀≤+−⇔ x x x mxmxx BC ( ) ' J J'I  ' trên x x xf + = .32 ' '  J'I  x x xf − =    −= = ⇔= − ⇔= ' '  J'I  ' '  lx lx x x xf [<- "J'  !  "R J\ !"  J'  =3&<3S0Z%[. KJ'' ≤⇔≤⇔ mm   U;0 K ≤ m P3Q>?7&, A.;+",-1L  J''I ' +−= mxxy M&2AX' ⇔ +)*5  = ý 237+]7 'J ^ xx 3&& ' 'J =− xx LTĐH - Chuyên Đề Khảo Sát Hàm Số _WWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam ( ) ( ) ( )    =−+ ∞+∪−∞−∈ ⇔      =−+ >− ⇔      =− >∆ ⇔ YY KK Y' IK Y  'J ' 'J 'J ' ' ' J ' ' 'J  xxxx m xxxx m xx ( ) ( ) ( ) ( ) φ ∈⇔         −= = ∞+∪−∞−∈ ⇔      =−       ∞+∪−∞−∈ ⇔ m m m m m m K K KK YYY I ' KK ' U;0?E2,&83P3>?7&, Ví dụ 4.%& mx mx y + + = `  3B,a?&<", B, ( ) ∞+ '  B, ( ) J −∞− Giải: 3.Bb- { } mRD −= c  ( ) ' '  ` mx m y + − = Ma?&<", mxy −≠∀>⇔    ( ) I I` ' −∈⇔>−⇔ mm U;0- ( ) I I −∈ m P3>?7&, .Bb- { } mRD −= c  ( ) ' '  ` mx m y + − = M ( ) ∞+ '  ( ) mxvàxy −≠∞+∈∀>⇔ '  LTĐH - Chuyên Đề Khảo Sát Hàm Số dWWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam  ( ) ( ) ( ) ( ) ( ) I ' II ' II ' ` ' >⇔    −≥ ∞+∪−∞−∈ ⇔    ≤− ∞+∪−∞−∈ ⇔    ∞+∉− >− ⇔ m m m m m m m U;0- I > m P3>?7&, .Bb- { } mRD −= c  ( ) ' '  ` mx m y + − = M ( ) J −∞− ( ) mxvàxy −≠−∞−∈∀<⇔ J    ( ) ( ) ( ) JI J I I J I I J ` ' ≤<−⇔    ≤ −∈ ⇔    −≥− −∈ ⇔    −∞−∉− <− ⇔ m m m m m m m U;0- JI ≤<− m P3>?7&, Ví dụ 5. (ĐỀ THI TUYỂN SINH ĐẠI HỌC KHỐI A, A1 NĂM 2013) %& I ' 0 " I" I" J J= − + + − ^F#3=.5 J?&< 9 ∞  Giải: .320e1RI" ' 9K"9I Zf&,⇔0e ( ) ^  x ≤ ∀ ∈ +∞ ' ' I K I    '   x x m x m x x x ⇔ − + + ≤ ∀ ∈ +∞ ⇔ ≤ − ∀ ∈ +∞ BC '   'g x x x= − F"T .32    ' 'g x x= −     Jg x x= ⇔ = LTĐH - Chuyên Đề Khảo Sát Hàm Số `WWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: Nguyễn Anh Tuấn – Trường THPT Nguyễn Thái Bình – Quảng Nam .32<- "J ∞+   "R9  ∞+  "  RJ =3&<3S0Z%[. Jm ⇔ ≤ − U;0 Jm ≤ −   +∞  BÀI TẬP TỰ LÀM J%& I ' I Yy x x mx= − − + + 2  C B, ?&< ( )  + ∞ bgh[iDMjkl4mn'` '%& I ' ' I' J K  J Jy x m x m m x = − + + + + 2%   .5?&< ( ) +∞ '  I(Dự bị 1 khối D 2003) %&- 2 2 x 5x m 4 y x 3 + + + = + ^J 3D<&,=oJ?1J .5J&?&< ( ) 1;+∞  2 .Ứng dụng tính đơn điệu để chứng minh bất đẳng thức: Ví dụ 1.%pX- 3"W"       ∈∀ '  π x        ∈∀< ' 3 π xxx  [ ] J J' 'Y −∈∀≤− xxx Giải: 3.32-"W"  >−⇔ xx BC xxxf  −= UF       ∈∀ '  π x .32       ∈∀≥=−= '  ' '&J ' π x x xxf  ' ' '  '         ∈=⇔=⇔=⇔=⇔= π ππ xDoxkxk xx xf LTĐH - Chuyên Đề Khảo Sát Hàm Số JWWW.ToanCapBa.Net . −∈∀≤−= xxxxf + CHUYÊN ĐỀ 2: CỰC TRỊ CỦA HÀM SỐ 1. Tìm điều kiện để hàm số đạt cực trị tai một điểm: LTĐH - Chuyên Đề Khảo Sát Hàm Số JJWWW.ToanCapBa.Net. '9JJ ∞+  LTĐH - Chuyên Đề Khảo Sát Hàm Số 'WWW.ToanCapBa.Net WWW.ToanCapBa.Net Giáo Viên: