GIO DC HNG PHC Ph Th, 09/2011 (C P N CHI TIT) GV: Lu Huy Thng GIO DC HNG PHC Chuyn luyn thi i hc khi A + B Tr s: Th trn Hng Sn _ Lm Thao _ Ph Th C s 2 : T X - Lm Thao - Ph Th C s 3 : Th trn Lm Thao - Lm Thao - Ph Th in thoi: 02106.259.638 g;, - -,;, ,-,_,[g -og;, -,[ ,[;,! Qg + + > = = >

> >

>s

> + s > s 222( 1) 2 3 2 0,1 2 013 2 0 1121 02 5 2 022( 1)(3 2) 0m x mx m xm mmm mmmmm mmm m m Cu 2.Cho hm sy x x mx3 23 4 = + (1) 1) Kho st s bin thin v v th ca hm s (1) khim 0 = . 2) Tm tt c cc gi tr ca tham s m hm s (1) ng bin trn khong( ; 0) . Gii - Tp xc nh: D = ;= + 2' 3 6 y x x m ,(1) ng bin trn khong (-;0) y > 0, x e (-;0) + >23 6 0 x x m x e (-;0) + >23 6 x x m x e (-;0) Xt hm s f(x) =+ 23 6 x x m trn (-;0] C f(x) = 6x + 6; f(x) = 0 x = -1 T bng bin thin: m 3 s Cu 3.Cho hm sy x m x mm x3 22 3( 2 1) 6 ( 1) 1 = + + + +c th (Cm). 1) Kho st s bin thin v v th ca hm s khi m = 0. 2) Tm m hm s ng bin trn khong(2; ) +Gii - Tp xc nh: D =y x m x m m2' 6 6(2 1) 6 ( 1) = + + +cm m m2 2(2 1) 4( ) 1 0 A = + + => x myx m' 01 == = + Ta c: y > 0, x (-;m) v (m + 1; +) Do : hm s ng bin trn(2; ) + m 1 2 + s m 1 s+--+-3 0 x f(x)x f(x) -+0-1 GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 2Cu 4.Cho hm s3 2(1 2 ) (2 ) 2 y x m x m x m = + + + + . 1) Kho st s bin thin v v th (C) ca hm s khi m = 1. 2) Tm m hm ng bin trn( ) 0; + . Gii -Tp xc nh: D ='= + +23 (1 2 ( 2 ) 2 ) y x mx m Hm ng bin trn(0; ) + y x mx m23 (1 2 ) (2 2 ) 0' + = + >vix 0 ) ( ; e + xf x mxx22 3( )4 12 + = >++ vix 0 ) ( ; e +Ta c: = +

+ = =+

'= =2222(2( ) 0 2(41)111)0 12xxxxxf x xx Lp bng bin thin ca hmf x ( )trn(0; ) + , t ta i n kt lun: | |> > |\ .1 52 4f m m Cu 5.Cho hm s 4 22 3 1 y x mx m = + (1),(m l tham s). 1) Kho st s bin thin v v th ca hm s (1) khi m = 1. 2) Tm m hm s (1) ng bin trn khong (1; 2). Gii -Tp xc nh: D =Ta c 3 2' 4 4 4 ( ) y x mx x x m = = +0 ms ,0,'> y x0 mstho mn. +0 m > ,0'= yc 3 nghim phn bit:,0, m m .Hm s (1) ng bin trn (1; 2) khi ch khi10 1 s < s m m . Vy( | ;1 me . Cu 6.Cho hm s mxyx m4 +=+ (1) 1) Kho st s bin thin v v th ca hm s (1) khim 1 = . 2) Tm tt c cc gi tr ca tham s m hm s (1) nghch bin trn khong( ;1) . Gii - Tp xc nh: D = R \ {m}. myx m224( )'=+. Hm s nghch bin trn tng khong xc nh y m 0 2 2'< < < (1) hm s (1) nghch bin trn khong( ;1) th ta phi cm m 1 1 > s (2) Kt hp (1) v (2) ta c:m 2 1 < s . GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 3Cu 7. Chngminhrng,hms 2si n cos y x x = + ngbintrnon 0;3t( ( vnghchbintrn on ;3t t ( ( Gii Hm s cho xc nh trn0; t( Ta c:' si n (2cos 1), (0; ) y x x x t = eV(0; ) si n 0 x x t e > nn trn 1(0; ) : ' 0 cos2 3y x xtt = = =+ Trn khong 0; : ' 03yt | |> |\ .nn hm s ng bin trn on0;3t( ( + Trn khong ; : ' 03yt t| |< |\ .nn hm s nghch bin trn on;3t t ( ( Cu 8. Cho hm s 3 23 y x x mx m = + + + . Tm m hm s nghch bin trn on c di bng 1Gii Hm s cho xc nh trnTa c: 2' 3 6 y x x m = + + c' 9 3m A = + Nu m > 3 th y > 0, x e , khi hm s ng bin trn, do m > 3 khng tha mn. + Nu m < 3, khi : y = 0 c hai nghim phn bit 1x , 2x1 2( ) x x < v hm s nghch bin trong on: 1 2; x x( vi di l = 2 1x x Theo Vi-t ta c: 1 2 1 22,3mx x xx + = =Hm s nghch bin trn on c di bng 1 l = 1 ( )222 1 1 2 1 24 91 ( ) 4 1 4 13 4x x x x xx m m = + = = =

GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 4PHN 2: CC TR CA HM S Cu 9.Cho hm sy x x mx m3 23 2 = + + +(m l tham s) c th l (Cm). 1) Kho st s bin thin v v th hm s khi m = 3. 2) Xc nh m (Cm) c cc im cc i v cc tiu nm v hai pha i vi trc honh. Gii - PT honh giao im ca (C) v trc honh:x x mx m3 23 2 0 (1) + + + = xgx x x m21( ) 2 2 0 (2) =

= + + = (Cm) c 2 im cc tr nm v 2 pha i vi trc 0xPT (1) c 3 nghim phn bit (2) c 2 nghim phn bit khc 1 mg m3 0( 1) 3 0A '= > = = m 3 < Cu 10. Cho hm sy x m x m m x3 2 2(2 1) ( 3 2) 4 = + + + (m l tham s) c th l (Cm). 1) Kho st s bin thin v v th hm s khi m = 1. 2) Xc nh m (Cm) c cc im cc i v cc tiu nm v hai pha ca trc tung. Gii -Tp xc nh: D =y x m x m m2 23 2(2 1) ( 3 2)'= + + + . (Cm)ccc imCvCT nmv hai phacatrctungPTy 0' = c 2 nghimtri du m m23( 3 2) 0 + > (*) Gi hai im cc tr l( ) ( )1 2 1 2; ; ; A B x y y xThc hin php chia y cho y' ta c: 1 1 2' 2 23 3 3 3m my x y x| | | | | |= + + |||\ . \ . \ . ( ) ( )1 1 1 2 2 22 22 2 ; 2 23 3 3 3| | | | | | | | + + + + ||||\ . \ . \ .= =.= =\y y x y ymxm m mx x Phng trnh ng thng i qua 2 im cc tr l A:22 23 3m my x| | | |= + + ||\ . \ .

Cc im cc tr cch u ng thngy x 1 = xy ra 1 trong 2 trng hp: TH1:ngthngiqua2imcctrsongsonghoctrngvingthngy x 1 = 2 32 13 2mm| | + =

= |\ . (tha mn) TH2: Trung im I ca AB nm trn ng thngy x 1 =

( ) ( )21 2 11 2 1222 2 1 12 22 23 32 23 .2 6 03 3| | | | + + + = + ||\ . \ .| | + = + + = = = |\ .I Ix m mx x x xxm myymyx Vy cc gi tr cn tm ca m l: 30;2m = ` ) Cu 13. Cho hm sy x mx m3 2 33 4 = +(m l tham s) c th l (Cm). 1) Kho st s bin thin v v th hm s khi m = 1. 2) Xc nh m (Cm) c cc im cc i v cc tiu i xng nhau qua ng thng y = x. Gii - Tp xc nh: D =Ta c:y x mx23 6' = ;xyx m002 =' = =. hm s c cc i v cc tiu th m = 0. th hm s c hai im cc tr l: A(0; 4m3), B(2m; 0) AB m m3(2 ; 4 ) = Trung im ca on AB l I(m; 2m3) A, B i xng nhau qua ng thng d: y = x AB dI d e m mm m332 4 02 == m22= GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 6Cu 14. Cho hm sy x mx m3 23 3 1 = + . 1) Kho st s bin thin v v th ca hm s khi m = 1. 2) Vi gi tr no ca m th th hm s c im cc i v im cc tiu i xng vi nhau qua ng thng d:x y 8 74 0 + = . Gii - Tp xc nh: D =y x mx23 6'= + ; y x x m 0 0 2'= = v = . Hm s c C, CT PTy 0'=c 2 nghim phn bit m 0 = . Khi 2 im cc tr l:A m B m m m3(0; 3 1), (2 ; 4 3 1) AB m m3(2 ; 4 ) Trung im I ca AB c to :I m m m3( ; 2 3 1) ng thng d:x y 8 74 0 + =c mt VTCP(8; 1) u = . A v B i xng vi nhau qua d I dAB de 38(2 3 1) 74 0. 0m m mABu + == m 2 = Cu 15. Cho hm sy x x mx3 23 = + (1). 1) Kho st s bin thin v v th ca hm s khi m = 0. 2) Vi gi tr no ca m th th hm s (1) c cc im cc i v im cc tiu i xng vi nhau qua ng thng d:x y 2 5 0 = . Gii - Tp xc nh: D =Ta cy x x mx y x x m3 2 23 ' 3 6 = + = +Hm s c cc i, cc tiu y 0'=c hai nghim phn bitm m 9 3 0 3 A' = > m ( ; 1 3) ( 1 3; ) e + +Ta c my x y m m x m21 12( 2 2) 4 13 3| | +'= + + + |\ . Gi s cc im cc i v cc tiu lAx y Bx y1 1 2 2( ; ), ( ; ) , I l trung im ca AB. y m m x m21 12( 2 2) 4 1 = + + + ; y m m x m22 22( 2 2) 4 1 = + + +v: x x mx x1 21 22( 1). 3 + = += Vy ng thng i qua hai im cc i v cc tiu ly m m x m22( 2 2) 4 1 = + + +A, B i xng qua (d):y x12= AB dI d e m 1 = . Cu 17. Cho hm sm x x m x y + + = 9 ) 1 ( 32 3, vim l tham s thc. 1) Kho st s bin thin v v th ca hm s cho ng vi1 = m . 2) Xc nhm hm s cho t cc tr ti 2 1, x xsao cho22 1s x x . Gii - Tp xc nh: D =Ta c. 9 ) 1 ( 6 3 '2+ + = x m x y+ Hm s t cc i, cc tiu ti 2 1, x x PT0 ' = yc hai nghim phn bit 2 1, x x PT0 3 ) 1 ( 22= + + x m xc hai nghim phn bit l2 1, x x .

> + = A 3 13 10 3 ) 1 ( '2mmm ) 1 (+ Theo nh l Viet ta c. 3 ); 1 ( 22 1 2 1= + = + x x m x xKhi : ( ) ( ) 4 12 1 4 4 4 222 122 1 2 1s + s + s m x x x x x x GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 8 m m2( 1) 4 3 1 + s s s (2) + T (1) v (2) suy ra gi tr ca m cn tm l3 1 3 < s mv. 1 3 1 s < + m Cu 18. Cho hm sy x mx mx m3 2(1 2 ) (2 ) 2 = + + + + , viml tham s thc. 1) Kho st s bin thin v v th ca hm s cho ng vi1 = m . 2) Xc nhm hm s cho t cc tr tix x1 2,sao chox x1 213 > . Gii - Tp xc nh: D =Ta c:y x mx m2' 3 (1 2 2 2 ) ( ) = + +Hm s c C, CTy ' 0 =c 2 nghim phn bitx x1 2,(gi sx x1 2< ) mm m m mm2 25' (1 2 ) 3(2 ) 4 5 041A

>

= = >

< (*) Hm s t cc tr ti cc imx x1 2, . Khi ta c: mx xmxx1 21 2(1 2 )3223 + = =

( ) ( )x x x x x x xx21 2 1 2 2 2 12113149 = + > > m m m m m m2 23 29 3 294(1 2 ) 4(2 ) 1 16 12 5 08 8+ > > > v v < Cu 19. Cho hm sy x m x m x3 21 1( 1) 3( 2)3 3= + + , vim l tham s thc. 1) Kho st s bin thin v v th ca hm s cho ng vim 2 = . 2) Xc nhm hm s cho t cc tr tix x1 2,sao chox x1 22 1 + = . Gii - Tp xc nh: D =Ta c:y x m x m22( 1) 3( 2)'= + Hm s c cc i v cc tiu y 0'= c hai nghim phn bitx x1 2,m m20 57 0 A' > + >(lun ng vi m) Khi ta c: x x mxx m1 21 22( 1)3( 2) + = = ( )x mx x m22 23 21 2 3( 2) = =

GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 9m m m24 348 16 9 04 + = = . Cu 20. Cho hm sy x mx x3 24 3 = + . 1) Kho st s bin thin v v th ca hm s khi m = 0. 2) Tm m hm s c hai im cc trx x1 2,thax x1 24 = . Gii - Tp xc nh: D =y x mx212 2 3'= + . Ta c:m m236 0, A' = + > hm s lun c 2 cc trx x1 2, . Khi :1 21 21 24614x xmx xx x = + = =

92m = Cu hi tng t: a)y x x mx3 23 1 = + + + ;x x1 22 3 + = S:m 105 = . Cu 21.Tm cc gi tr ca m hm s 3 2 21 1( 3)3 2y x mx m x = + c cc i 1x , cc tiu 2xng thi 1x ; 2xl di cc cnh gc vung ca mt tam gic vung c di cnh huyn bng 52 Gii Cch 1: Min xc nh: D = c 2 2 2 2' 3; ' 0 0 y x mx m y x mx m = + = + =Hm s t cc i ti 1xcc tiu ti 2xtha mn yu cu bi ton khi v ch khi pt y= 0 c hai nghim dng phn bit, trit tiu v i du qua hai nghim :220 4 0 2 20 0 0 3 2 (* )03 3 3 0m mS m m mPm m m A > > < < > > > < < >< v > > Theo Vi-t ta c: 1 221 23x x mxx m + == M 2 2 2 2 21 2 1 2 1 25 142( ) 4 5 5 2 4( 3) 52 2x x x x xx m m m + = + = = = = i chiu iu kin (*) ta c: 142m =

Cu 22. Cho hm s 3 2 21( 1) 1 (Cooth(C ))3my x mx m x = + + GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 101) Kho st s bin thin v v th hm s vi m = 2 2) Tm m hm s c cc i cc tiu v: D2C CTy y + >Gii 2 22 2'Ta co: ' 2 ( 1)' 1 1 01' 01yy x mx mm mx myx m= + A = + => = + = = ( 1) ( 1) CD CT m my y y y+ + = +3 32 2 2 23 2( 1) ( 1)[ ( 1) ( 1)( 1) 1] [ ( 1) ( 1)( 1) 1]3 31 02 2 2 2 ( 1) 011 0KL: 1m mmm m m mm m mmm m mmmmm+ = + + + + + + + < > > < Cu 23. Cho hm s y m x x mx3 2( 2) 3 5 = + + + , m l tham s. 1) Kho st s bin thin v v th (C) ca hm s khi m = 0. 2) Tm cc gi tr ca m cc im cc i, cc tiu ca th hm s cho c honh l cc s dng. Gii - Cc im cc i, cc tiu ca th hm s cho c honh l cc s dng PTy m x x m =2' 3( 2) 6 0 = + + +c 2 nghim dng phn bit a mmmm m mmm m m Pmm mSm2( 2) 0' 9 3 ( 2) 0' 2 3 0 3 10 0 3 2 03( 2)2 0 2302AA = + == + > = + > < < < < < < = > + + < < = >+ Cu 24. Cho hm sy x x3 2 3 2 = + (1) 1) Kho st s bin thin v v th ca hm s (1). 2) Tm imM thuc ng thng d:y x 3 2 = sao tngkhong cch t M ti hai im cc tr nh nht. Gii - Cc im cc tr l: A(0; 2), B(2; 2). Xt biu thcgxy x y ( , ) 3 2 = ta c: A A A A B B B Bgx y x y gx y x y ( , ) 3 2 4 0; ( , ) 3 2 6 0 = = < = = > 2 im cc i v cc tiu nm v hai pha ca ng thng d:y x 3 2 = . Do MA + MB nh nht 3 im A, M, B thng hng M l giao im ca d v AB. GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 11 Phng trnh ng thng AB:y x 2 2 = +Ta im M l nghim ca h: 43 252 2 25xy xy xy == = += 4 2;5 5M | | |\ . Cu 25. Cho hm sy x mx mx m3 2(1 2 ) (2 ) 2 = + + + + (m l tham s) (1). 1) Kho st s bin thin v v th hm s (1) khi m = 2. 2) Tm cc gi tr ca m th hm s (1) c im cc i, im cc tiu, ng thi honh ca im cc tiu nh hn 1. Gii - Tp xc nh: D =y x mx m gx23 2(1 2 ) 2 ( )'= + + =YCBT phng trnhy 0'=c hai nghim phn bitx x1 2, tha mn:x x1 21 < < . m mg mS m24 5 0(1) 5 7 02 112 3A ' = >= + >= < m5 74 5< < . Cu 26. Cho hm s 3 2 2 33 3( 1) y x mx m x m m = + +(1) 1) Kho st s bin thin v v th ca hm s (1) khi m = 1. 2) Tm m hm s (1) c cc tr ng thi khong cch t im cc i ca th hm s n gc ta O bng2ln khong cch t im cc tiu ca th hm s n gc ta O. Gii - Tp xc nh: D =Ta c 2 23 6 3( 1)'= + y x mx mHm s (1) c cc tr th PT0'= y c 2 nghim phn bit 2 22 1 0 x mx m + =c 2 nhim phn bit1 0, m A = > Khi : im cc i Am m ( 1; 2 2 ) v im cc tiuBm m ( 1; 2 2 ) + Ta c23 2 22 6 1 03 2 2mOA OB m mm

= += + + = = . Cu 27. Cho hm sy x mx m x m m3 2 2 3 23 3(1 ) = + + + (1) 1) Kho st s bin thin v v th ca hm s (1) khim 1 = . 2) Vit phng trnh ng thng qua hai im cc tr ca th hm s (1). Gii - Tp xc nh: D = GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 12y x mx m2 23 6 3(1 )'= + + .PTy 0'=cm 1 0, A => th hm s (1) lun c 2 im cc trx y x y1 1 2 2( ; ), ( ; ) . Chia y cho y' ta c: my x y x m m2123 3| |'= + + |\ . Khi :y x m m21 12 = + ; y x m m22 22 = +PT ng thng qua hai im cc tr ca th hm s (1) ly x m m22 = + . Cu 28. Cho hm s 3 23 2 y x x mx = + c th l (Cm). 1) Kho st s bin thin v v th ca hm s khi m = 1. 2) Tm m (Cm) c cc im cc i, cc tiu v ng thng i qua cc im cc tr song song vi ng thng d:y x 4 3 = + . Gii - Tp xc nh: D =Ta c: 2' 3 6 = y x x m. Hm s c C, CT 2' 3 6 0 y x x m = =c 2 nghim phn bit 1 2; x x' 9 3 0 3 m m A = + > > (*) Gi hai im cc tr l( ) ( )1 2 1 2; ; ; A B x y y xThc hin php chia y cho y' ta c: 1 1 2' 2 23 3 3 3m my x y x| | | | | |= + + |||\ . \ . \ . ( ) ( )1 1 1 2 2 22 22 2 ; 2 23 3 3 3| | | | | | | | + + + + ||||\ . \ . \ .= =.= =\y y x y ymxm m mx x Phng trnh ng thng i qua 2 im cc tr l d:22 23 3m my x| | | |= + + ||\ . \ . ng thng i qua cc im cc tr song song vi d:y x 4 3 = + 22 4332 33mmm | | + = | \ . =| | = |\ . (tha mn) Cu 29. Cho hm s 3 23 2 y x x mx = + c th l (Cm). 1) Kho st s bin thin v v th ca hm s khi m = 1. 2) Tm m (Cm) c cc im cc i, cctiu v ng thng i qua cc im cc tr to vi ng thng d:x y 4 5 0 + =mt gc 045 . Gii - Tp xc nh: D =Ta c: 2' 3 6 = y x x m. GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 13 Hm s c C, CT 2' 3 6 0 y x x m = =c 2 nghim phn bit 1 2; x x' 9 3 0 3 m m A = + > > (*) Gi hai im cc tr l( ) ( )1 2 1 2; ; ; A B x y y xThc hin php chia y cho y' ta c: 1 1 2' 2 23 3 3 3m my x y x| | | | | |= + + |||\ . \ . \ . ( ) ( )1 1 1 2 2 22 22 2 ; 2 23 3 3 3| | | | | | | | + + + + ||||\ . \ . \ .= =.= =\y y x y ymxm m mx x Phng trnh ng thng i qua 2 im cc tr l A:22 23 3m my x| | | |= + + ||\ . \ . t 223mk| |= + |\ .. ng thng d:x y 4 5 0 + =c h s gc bng 14 . Ta c: 3 39 1 1115 10 4 44tan 4511 1 5 11144 4 3 2k m k kkkk k k m

= = + = +

=

+ = + = =

Kt hp iu kin (*), suy ra gi tr m cn tm l: 12m = Cu 30. Cho hm s y x x m3 23 = + + (1) 1) Kho st s bin thin v v th ca hm s (1) khim 4 = . 2) Xc nh m th ca hm s (1) c hai im cc trA, B sao cho AOB0120 = . Gii - Tp xc nh: D =Ta c:y x x23 6'= + ;x y myxy m2 400 = = +'= = = Vy hm s c hai im cc tr A(0 ; m) v B(2 ; m + 4) OA mOB m (0; ), ( 2; 4) = = + . AOB0120 = thAOB1cos2= ( )( )m mmm m mmm mm m2 222 24 0 ( 4) 14 ( 4) 2 ( 4)2 3 24 44 04 ( 4) < < + = + + = + + + =+ + mmm4 012 2 312 2 333 < < + = = Cu 31. Cho hm s y x mx m x m3 2 2 3 3 3( 1) = + (Cm) 1) Kho st s bin thin v v th ca hm s (1) khim 2 = . 2) Chngminhrng(Cm)lunc imccivimcctiulnltchytrnming thng c nh. GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 14Gii - Tp xc nh: D =y x mx m2 23 6 3( 1)'= + ; x myx m101 = +'= = im cc iMm m ( 1; 2 3 )chy trn ng thng c nh: 12 3x ty t= + = im cc tiuNm m ( 1; 2 ) + chy trn ng thng c nh:12 3x ty t= + = Cu 32. Cho hm s 3 2 3 23( 1) 3 ( 2) 3 y x m x mm x m m = + + + + + + 1) Kho st s bin thin v v th hm s vi m = 0. 2) Chng minh rng vi mi m hm s lun c 2 cc tr v khong cch gia hai im ny khng ph thuc vo v tr ca m. Gii Ta c: 22' 3 6( 1) 6 ( 2); ' 0x my x m x mm yx m = = + + + + = = Hm s ng bin trn cc khong (-;-2 - m) v (-m;+), nghch bin trn khong (-2 - m;-m) v2 ; 4; ; 0CD CD CT CTx my x my = = = =Khi , khong cch gia hai im cc tr l: 2 2( 2 ) (4 0) 2 5 m m + + = iu phi chng minh. Cu 33. Cho hm s2 32 3+ = mx x x y (1) vi m l tham s thc. 1) Kho st s bin thin v v th ca hm s (1) khi m = 0. 2) nh m hm s (1) c cc tr, ng thi ng thng i qua hai im cc tr ca th hm s to vi hai trc ta mt tam gic cn. Gii Ta c: 2' 3 6 y x x m = Hm s c cc tr khi v ch khiy = 0 c 2 nghim phn bit ' 9 3 0 3 m m A = + > > (1) Ly y chia cho y ta c:3 21 23 2 ( 1). ' ( 2) 23 3 3m my x x mx x y x = + = + + ng thng qua hai im cc tr ca th hm s c phng trnh 32 ) 232(mxmy + =ng thng ny ct 2 trc Ox v Oy ln lt tai|.|

\| ||.|

\|+36; 0 , 0 ;) 3 ( 26 mBmmA GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 15 Tam gic OAB cn khi v ch khiOA OB =6 62( 3) 3 =+m mm9 36; ;2 2 = = = m m mVi m = 6th O B A do so vi iu kin ta nhn 23 = m Cu 34. Cho hm sy x mx4 21 32 2= +(1) 1) Kho st s bin thin v v th ca hm s (1) khim 3 = . 2) Xc nh m th ca hm s (1) c cc tiu m khng c cc i. Gii - Tp xc nh: D =y x mx xx m3 22 2 2 ( )'= = .xyx m200 ='= = th ca hm s (1) c cc tiu m khng c cc i PTy 0'=c 1 nghim m 0 s Cu 35. Cho hm s 4 22 4 ( )my x mx C = + 1) Kho st s bin thin v v th hm s vi m = 2 2) Tm cc gi tr ca m tt c cc im cc tr ca( )mCu nm trn cc trc ta . Gii Ta c: 320' 4 4 ; ' 0xy x mx yx m == + = = Nu m s 0 th hm s c 1 im cc tr duy nht v im nm trn trc tung. Nu m > 0 th th hm s khi c 3 im cc tr. Mt im cc tr nm trn trc tung v hai im cc tr cn li c ta : 2( ; 4) mm Cc im ny ch c th nm trn trc honh. iu kin cc im nm trn trc honh l 204 0mm > = m = 2 Kt lun: 20mm =

s Cu 36. Cho hm s 4 2 2( ) 2( 2) 5 5 = = + + + y f x x m x m mmC ( ) . 1) Kho st s bin thin v v th(C) hm s khim = 1. 2) Tm cc gi tr ca m th mC ( )ca hm s c cc im cc i, cc tiu to thnh 1 tam gic vung cn. Gii - Tp xc nh: D = GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 16Ta c( )3204 4( 2) 02= '= + = = xf x x m xx m Hm s c C, CT PTf x ( ) 0'=c 3 nghim phn bit m 2 < (*) Khi to cc im cc tr l: ( ) ( ) ( ) A m m B m m C m m20; 5 5, 2 ;1 , 2 ;1 + ( ) ( )AB m m m AC m m m2 22 ; 4 4, 2 ; 4 4 = + = + Do AABC lun cn ti A, nn bi ton tho mn khi AABC vung ti A( ) 1 1 2 0 .3= = = m m AC AB (tho (*)) Cu 37.Cho hm s( )mC m m x m x y 5 5 ) 2 ( 22 2 4+ + + =1) Kho st s bin thin v v th hm s khi m = 1. 2) Vinhnggi trno cam th th (Cm) c im cc iv imcc tiu, ng thi cc im cc i v im cc tiu lp thnh mt tam gic u. Gii - Tp xc nh: D =Ta c( )3204 4( 2) 02= '= + = = xf x x m xx m Hm s c C, CT PTf x ( ) 0'=c 3 nghim phn bit m 2 > < = = = Khi : 1 1 2 2( ; 1); ( ; 1). Bx mx C x mx + + V trung im ca AC nn 2 12 (1) x x =M 1x , 2xl nghim ca phng trnh: 22 6 0 x x m + =nn 1 21 23(2)2x xmxx + == T (1) v (2) m = 4 Cu 49. Cho hm sy x mx m x m3 2 2 23 3( 1) ( 1) = + ( m l tham s)(1). 1) Kho st s bin thin v v th ca hm s (1) khim 0. =2) Tm cc gi tr ca m th hm s (1) ct trc honh ti 3 im phn bit c honh dng. - THS (1) ct trc honh ti 3 im phn bit c honh dng, ta phi c: C CTC CTco cc tr y yx xa y(1) 2. 00, 0. (0) 0< > >< (*)Trong : +y x mx m x m3 2 2 23 3( 1) ( 1) = + y x mx m2 23 6 3( 1)' = + + ym m m2 21 0 0, A' = + = > + CCTx m xyx m x101 = ='= = + = Suy ra: (*) mmmm m m mm2 2 221 01 03 1 2( 1)( 3)( 2 1) 0( 1) 0 >+ > < < + < < Cu 50. Cho hm s 3 21 23 3y x mx x m = + +c th mC ( ) . 1) Kho st s bin thin v v th ca hm s khi m = 1. 2) Tm m mC ( ) ct trc honh ti 3 im phn bit c tng bnh phng cc honh ln hn 15. Gii - YCBT x mx x m3 21 203 3 + + =(*) c 3 nghim phn bit thax x x2 2 21 2 315 + + > . Ta c: (*)x x mx m2( 1)( (1 3 ) 2 3 ) 0 + = xgx x mx m21( ) (1 3 ) 2 3 0 =

= + = GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 23 Do : YCBT gx ( ) 0 =c 2 nghimx x1 2,phn bit khc 1 v thax x2 21 214 + > . m 1 >Cu hi tng t i vi hm s: 3 23 3 3 2 y x mx x m = + + Cu 51. Cho hm sm x x x y + = 9 32 3, trong m l tham s thc. 1) Kho st s bin thin v v th (C) ca hm s cho khi0 = m . 2) Tm tt c cc gi tr ca tham sm thhm s choct trc honh ti 3 im phn bit c honh lp thnh cp s cng. Gii - th hm s ct trc honh ti 3 im phn bit c honh lp thnh cp s cng Phng trnh 3 23 9 0 + = x x x mc 3 nghim phn bit lp thnh cp s cng Phng trnh 3 23 9 x x x m = c 3 nghim phn bit lp thnh cp s cng ng thngy m = i qua im un ca th (C) . 11 11 m m = = Cu 52. Cho hm sy x mx x3 23 9 7 = + c th (Cm), trong m l tham s thc. 1) Kho st s bin thin v v th (C) ca hm s cho khi0 = m . 2) Tmm (Cm) ct trc honh ti 3 im phn bit c honh lp thnh cp s cng. Gii - Honh cc giao im l nghim ca phng trnh:x mx x3 23 9 7 0 + =(1) Gi honh cc giao im ln lt l x x x1 2 3; ;ta c:x x x m1 2 33 + + =x x x1 2 3; ;lp thnh cp s cng thx m2 =l nghim ca phng trnh (1) m m32 9 7 0 + = mmm11 1521 152

=

+

=

=

Th li ta cm1 152 =l gi tr cn tm. Cu 53. Cho hm s 3 23 y x mx mx = c th (Cm), trong m l tham s thc. 1) Kho st s bin thin v v th (C) ca hm s cho khim 1 = . 2) Tmm (Cm) ct ng thng d:y x 2 = +ti 3 im phn bit c honh lp thnh cp s nhn. Gii - Xt phng trnh honh giao im ca (Cm) v d: ( ) ( )3 2 3 23 2 3 1 2 0 x mx mx x g x x mx m x = + = + = GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 24kcn:Gis(C)ct dti 3 imphn bitc honh 1 2 3; ; x x xlnltlpthnhcps nhn. Khi ta c:( ) ( )( )( )1 2 3g x x x x x x x = Suy ra: 1 2 31 2 2 3 1 31 2 3312x x x mx x x x x x mx x x+ + = + + = = V 2 3 31 3 2 2 22 2 x x x x x = = =nn ta c: 3351 4 2.33 2 1m m m = + = + k : Vi 353 2 1m = +, thay vo tnh nghim thy tha mn. Vy 353 2 1m = + Cu 54. Cho hm s: y = 2x3 - 3x2 + 1 (1) 1) Kho st s bin thin v v th (C) cahm s (1) 2) Tm trn (C) nhng im M sao cho tip tuyn ca (C) ti M ct trc tung ti im c tung bng 8. Gii Gi s M (x0; y0) e(C)y0 = 2x03 - 3x02 + 1 Ta c : 2' 3 6 y x x = Tip tuyn ( A) ca (C) ti M: y = (6x02 - 6x0) (x - x0) + 2x03 - 3x02 + 1 ( A) i qua im P(0 ; 8) 8 = -4x03 + 3x02 + 1 (x0 + 1) (4x02 - 7x0 + 7) = 0 x0 = -1 ; (4x02 - 7x0 + 7 > 0,x0) Vy, c duy nht im M (-1 ; -4) cn tm. Cu 55. Cho hm s y x mx m x3 22 ( 3) 4 = + + + +c th l (Cm) (m l tham s). 1) Kho st s bin thin v v th (C1) ca hm s trn khi m = 1. 2) Cho ng thng (d):y x 4 = +v im K(1; 3). Tm cc gi tr ca m (d) ct (Cm) ti ba im phn bit A(0; 4), B, C sao cho tam gic KBC c din tch bng8 2 . Gii - Phng trnh honh giao im ca (Cm) v d l: x mx m x x xx mx m3 2 22 ( 3) 4 4 ( 2 2) 0 + + + + = + + + + = x ygx x mx m 20 ( 4)( ) 2 2 0 (1) = = = + + + = (d) ct (Cm) ti ba im phn bit A(0; 4), B, C(2) c 2 nghim phn bit khc 0. m mm mmg m/ 21 22 02(0) 2 0A s v >= > = = + =(*) GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 25 Khi : B C B Cx x mx x m 2 ; . 2 + = = + . Mt khc:dKd1 3 4( , ) 22 += = .Do : KBCS BC dKd BC BC218 2 . ( , ) 8 2 16 2562A= = = = B C B Cx x y y2 2( ) ( ) 256 + =B C B Cx x x x2 2( ) (( 4) ( 4)) 256 + + + = B C B C B Cx x x x x x2 22( ) 256 ( ) 4 128 = + =m m m m m2 21 1374 4( 2) 128 34 02 + = = = (tha (*)).Vym1 1372= . Cu 56. Cho hm sy x x3 23 4 = +c th l (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Gi kdl ng thng i qua imA( 1; 0) vi h s gck k ( ) e . Tmk ng thng kdct th (C) ti ba im phn bit A, B, C v 2 giao imB, C cng vi gc to Oto thnh mt tam gic c din tch bng 1. Gii - Ta c: kd y kx k : = +kx y k 0 + =Phng trnh honh giao im ca (Cm) v d l:x x kx k x x k x3 2 23 4 ( 1) ( 2) 0 1 ( + = + + = = hocx k2( 2) = kdct (C) ti 3 im phn bit kk09 > = Khi cc giao im l( ) ( ) A B k k k k C k k k k ( 1; 0), 2 ; 3 , 2 ; 3 + + . kkBC k k dOBC dOdk222 1 , ( , ) ( , )1= + = =+ OBCkS k k k k k kk2 321. .2 . 1 1 1 1 121A= + = = ==+ Cu 57. Cho hm sy x x3 23 2 = +c th l (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Gi E l tm i xng ca th (C). Vit phng trnh ng thng qua E v ct (C) ti ba im E, A, B phn bit sao cho din tch tam gic OAB bng2 . Gii - Ta c: E(1; 0). PT ng thng A qua E c dngy kx ( 1) = . PT honh giao im ca (C) v A:x x x k2( 1)( 2 2 ) 0 = GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 26A ct (C) ti 3 im phn bit PTx x k22 2 0 =c hai nghim phn bit khc 1k 3 > OABS dO AB k k1( , ). 32A= A = +k k 3 2 + = kk11 3 =

= Vy c 3 ng thng tho YCBT: ( )y x y x 1; 1 3( 1) = + = . Cu 58. Cho hm sy x mx32 = + +c th (Cm) 1) Kho st s bin thin v v th ca hm s khi m = 3. 2) Tm m th (Cm) ct trc honh ti mt im duy nht. Gii - Phng trnh honh giao im ca (Cm) vi trc honh: x mx32 0 + + = m x xx22( 0) = =Xt hm s: xf x x f x xxx x322 22 2 2 2( ) '( ) 2 += = + = Ta c bng bin thin: f x ( )'f x ( ) ++ th (Cm) ct trc honh ti mt im duy nhtm 3 > . Cu 59. Cho hm sy x m x mx3 22 3( 1) 6 2 = + + c th (Cm) 1) Kho st s bin thin v v th ca hm s khi m = 1. 2) Tm m th (Cm) ct trc honh ti mt im duy nht. Gii - Tp xc nh: D == + +2' 6 6( 1) 6 y x m x mA = + = 2 2'' 9( 1) 36 9( 1)ym m m Th1: m = 1 hm s ng bin trn th hm s ct trc honh ti 1 im duy nht. m = 1(tha mn) Th2:m 1 Hm s c cc i v cc tiu. Gi 1x , 2x l cc im cc tr ca hm s 1x , 2xl cc nghim ca phng trnh y = 0 Theo Viet ta c: + = +=1 21 21.x x mx x m

Ly y chia cho y ta c: += + +21( ) ' ( 1) 2 ( 1)3 6x my y m x mm Phng trnh i qua im cc i v cc tiu ca hm s GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 27 = + +2( 1) 2 ( 1) y m x mm hm s ct trc honh ti 1 im duy nht < . 0CD CTy y + + + + > + + + + > + + + + > + + + + > + + + + + + >2 21 24 2 2 2 21 2 1 24 2 2 2 22 2 2 23 2 3 2 2[ ( 1) 2 ( 1)] [ ( 1) 2 ( 1)] 0( 1) ( 1) ( 2)( ) ( 2) 0( 1) ( 1) ( 2)( 1) ( 2) 0( 1) [( 1) ( 2)( 1) ( 2) ] 02 2 2 4 4 0( m x mm m x mmm xx m m m x x m mm m m m m m m mm m m m m m mm m m m m m m m V m = < < < +21)2 2 01 3 1 3m mm Kt lun: < < + 1 3 1 3 m Cu 60. Cho hm sy x x x3 26 9 6 = + c th l (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) nh m ng thngd y mx m ( ) : 2 4 = ct th (C) ti ba im phn bit. Gii - PT honh giao im ca (C) v (d):x x x mx m3 26 9 6 2 4 + = x x x m2( 2)( 4 1 ) 0 + = xgx x x m22( ) 4 1 0 =

= + = (d) ct (C) ti ba im phn bit PTgx ( ) 0 =c 2 nghim phn bit khc 2 m 3 > Cu 61. Cho hm sy x x3 2 3 1 = + . 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm m ng thng (A):y m x m (2 1) 4 1 = ct th (C) ti ng hai im phn bit. Gii - Phng trnh honh giao ca (C) v (A):x x m x m3 2 3 (2 1) 4 2 0 + + =x x x m2( 2)( 2 1) 0 =xf x x x m22( ) 2 1 0 (1) = = = (A) ct (C) ti ng 2 im phn bit (1) phi c nghimx x1 2,tha mn:x xx x1 21 222= =

= =

baf0220(2) 0AA =

=

>

= mmm8 5 01228 5 02 1 0 + =

=

+ >

+ = mm5812

=

= Vy:m58= ;m12= . GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 28Cu 62. Cho hm s 3 23 2 y x m x m = + c th (Cm). 1) Kho st s bin thin v v th ca hm s khi m = 1. 2) Tm m th (Cm) ct trc honh ti ng hai im phn bit. Gii - (Cm) ct trc honh ti ng hai im phn bit th (Cm) phi c 2 im cc tr 0'= yc 2 nghim phn bit 2 23 3 0 x m =c 2 nghim phn bit 0 m =Khi ' 0 y x m = = . (Cm) ct Ox ti ng 2 im phn bityC = 0 hoc yCT = 0 Ta c: + 3( ) 0 2 2 0 0 y m m m m = + = =(loi) + 3( ) 0 2 2 0 0 1 y m m m m m = + = = v = Vy:1 m = Cu 63. Cho hm s 3 26 9 (1) y x x x = +1) Kho st s bin thin v v th (C) ca hm s (1) 2) Tm m ng thng (d): y mx = ct (C) ti ba im O (0;0), A v B. Chng t rngkhi m thay i, trung im I ca on AB lun nm trn mt ng thng song song vi Oy. Gii Phng trnh honh giao im ca ng thng (d) vi th (C) l:3 2206 9 (1)6 9 0 (2)xx x x mxx x m = + = + = (d) ct (C) ti 3 im phn bit O(0;0), A, B (1) c 3 nghim phn bit (2) c 2 nghim phn bit khc x= 0 ' 00 9 (* )9 0mmA >< = = Vi iu kin (*), A, B l 2 im c honh ln lt l;A Bx x l 2 nghim ca phng trnh (2) I l trung im ca on thng AB nn honh I:32A BIx xx+= =I eA c phng trnh l x = 3,A song song vi Oy khi m thay i(0 9) m < = Cu 64. Cho hm s 3 23 ( 1) 1 y x mx m x m = + + + c th l( )mC1) Kho st s bin thin v v th 1( ) Ckhi m = 1 2) Tm tt c cc gi tr ca m d:2 1 y x m = ct th( )mC ti ba im phn bit c honh ln hn hoc bng 1. Gii Phng trnh honh giao im ca( )mCvi ng thng (d): GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 29 3 23 2223 ( 1) 1 2 13 ( 3) 2 2 0( 1) (1 3 ) 2 2 0 (1)1(1 3 ) 2 2 0 (2)x mx m x m x mx mx m x mx x mx mxx mx m + + + = + + + = ( + = = + = ( )mCct (d) ti 3 im phn bit c honh ln hn hoc bng 1 (1) c 3 nghim phn bit ln hn hoc bng 1 (2) c hai nghim phn bit ln hn 1. Xt phng trnh (2); Ta c: 2 2(1 3 ) 8 8 9 2 9 0, m m m m m A = + + = + + > m (2) lun c 2 nghim phn bit 1 2; x x(2) c 2 nghim ln hn 1 1 2 1 21 0 1 1 x x x x < < < < t t = x - 1 x = t + 1 (2) 2( 1) (1 3 )( 1) 2 2 0 t m t m + + + = 23(1 ) 5 0 (3) t mt m + =(2) c 2 nghim tha mn:1 21 x x < < (3) c 2 nghim dng phn bit: 03( 1) 05 0S m vnp mA >= > = > Kt lun: khng c gi tr m Cu 65. Cho hm s 33 2 y x x = +1) Kho st s bin thin v v th hm s. 2) Vit phng trnh ng thng ct th (C) ti 3 im phn bit A, B, C sao cho2Ax =v 2 2 BC =Gii Vi2 4A Ax y = =Phng trnh ng thng (d) i qua im A(2;4) l:( ) : ( 2) 4A Ay k x x y d y k x = + = +Phng trnh honh giao im ca (C) vi ng thng (d) 3 223 2 ( 2) 4 ( 2)( 2 1) 02( ) 2 1x x k x x x x kxg x x x k + = + + + = = = + + iu kin c BC: ' 0 0(2) 0 9kg k A > >

= = Khi . Ta ca 1 1 2 2( ; ); ( ; ) Bx y Cx y tha mn h phng trnh: 22 1 0 (1)2 4 (2)x x ky kx k+ + == + GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 30Ta c: 2 12 1 2 1(1) 2 ' 2(2) ( 2x x ky y k x x k k = A = = = Theo gi thit ta c: BC =2 2 3 34 4 2 2 4 4 8 0 1 k k k k k + = + = =Vy:: 2 d y x = + Cu 66. Cho hm s 3 24 6 1 y x mx = +(C), m l tham s. Tm m ng thng d: y = -x + 1 ct th hm s ti 3 im A(0;1), B, C vi B, C i xng nhau qua ng phn gic th nht. Gii Giao ca (C) v (d) c honh l nghim ca phng trnh: 3 2 24 6 1 1 (4 6 1) 0 x mx x x x mx + = + + = phng trnh c 3 nghim phn bit th 24 6 1 0 x mx + = c hai nghim phn bit. 22 2' 9 4 0 ;3 3m m mA = > > 0) (1) = + = = 2 11 01tt mt mt m th hm s ct trc honh ti 4 im phn bit (1) c 4 nghim phn bit 0 < m - 1= 1 mm12 >= Cu 68. Cho hm s( )4 22 1 2 1 y x m x m = + + +c th l ( )mC . 1) Kho st s bin thin v v th ca hm s cho khi0 = m . 2) nhm th ( )mCct trc honh ti 4 im phn bit c honh lp thnh cp s cng. Gii -Xt phng trnh honh giao im:( )4 22 1 2 1 0 x m x m + + + =(1) GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 31 t 2, 0 t x t = >th (1) tr thnh:( )2( ) 2 1 2 1 0 f t t m t m = + + + = . (Cm) ct Ox ti 4 im phn bit thf t ( ) 0 =phi c 2 nghim dng phn bit ( )2' 012 1 0 202 1 0mmS mmP mA = >> = + > = = + > (*) Vi (*), gi 1 2t t y = 6m + 2 => th hm s nhn im U(1; 6m+2) lm im un im un thuc Ox khi yU = 0 6m+2 = 0 13m=Vy 13m=l gi tr cn tm Cu 72. Cho hm s ( )4 22 1 2 1 y x m x m = + + +c th l (Cm), m l tham s. 1) Kho st s bin thin v v th ca hm s khi m = 0. 2) Tm m th (Cm) ct trc honh ti 3 im phn bit u c honh nh hn 3. Gii - Xt phng trnh honh giao im:( )4 22 1 2 1 0 x m x m + + + = (1) t 2, 0 t x t = >th (1) tr thnh:( )2( ) 2 1 2 1 0 f t t m t m = + + + = . (Cm) ct Ox ti 3 im phn bit c honh nh hn 3 ( ) f t c 2 nghim phn bit 1 2, t tsao cho: 1 21 20 30 3t tt t= < <

< < s ( )( )( )22' 0' 03 4 4 01(0) 2 1 0 122 1 02 1 32 1 0A = >A = > = s = + = = v > = + > = + mmf mf m m mS mS mP m Vy:112m m = v > . GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 33 Cu 73. Cho hm s 4 2 2 42 2 y x m x m m = + +(1), vi m l tham s. 1) Kho st s bin thin v v th ca hm s (1) khi1 m = .. 2) Chng minh th hm s (1) lun ct trc Ox ti t nht hai im phn bit, vi mi0 m < . Gii - Phng trnh honh giao im ca th (1) v trc Ox: 4 2 2 42 2 0 x m x m m + + = (1) t( )20 t x t = > , (1) tr thnh : 2 2 42 2 0 t m t m m + + = (2) Ta c :' 2 0 m A = >v 22 0 S m = >vi mi0 m > . Nn (2) c nghim dng (1)ct nht 2 nghimphn bit th hms(1)luncttrcOxtit nht hai im phn bit. Cu 74. Cho hm s xyx2 12+=+ c th l (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Chngminh rng ng thng d:y x m = +lun ct th (C) ti hai im phn bit A, B. Tm m on AB c di nh nht. Gii - PT honh giao im ca (C) v d: xx mx2 12+= ++

xf x x mx m22( ) (4 ) 1 2 0 (1) = = + + = Do (1) cm21 0 A = + >vf m m m2( 2) ( 2) (4 ).( 2) 1 2 3 0, = + + = = nn ng thng d lun lun ct th (C ) ti hai im phn bit A, B. Ta c: A A B By m x y m x ; = = nn B A B AAB x x y y m2 2 2 2( ) ( ) 2( 12) = + = +Suy ra AB ngn nht AB2 nh nht m 0 = . Khi :AB 24 = . Cu hi tng t i vi hm s:a) 21xyx= S: m = 2b) xyx12= S:m12=

Cu 75. Cho hm s 31xyx=+. 1) Kho st s bin thin v v th (C) ca hm s. 2) Vit phng trnh ng thng d qua im( 1;1) Iv ct th (C) ti hai im M, Nsao cho Il trung im ca on MN. Gii - Phng trnh ng thng( ) : 1 1 d y k x = + +d ct (C) ti 2 im phn bit M, N 311 = + ++xkx kx c 2 nghim phn bit khc1 . GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 34 2( ) 2 4 0 = + + + = f x kx kx kc 2 nghim phn bit khc1 04 0 0( 1) 4 0= A = > < = =kk kf Mt khc:2 2M N Ix x x + = = I l trung im MN vi0 k < . Kt lun: Phng trnh ng thng cn tm l1 y kx k = + +vi0 k < . Cu 76. Cho hm s 2 41xyx+= (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Gi (d) l ng thng qua A(1; 1) v c h sgc k. Tm k (d) ct (C) ti hai im M, N sao cho3 10 MN = . - Phng trnh ng thng( ) : ( 1) 1. d y k x = +Biton trthnh:Tmk h phngtrnhsauc hai nghim 1 1 2 2( ; ), ( ; ) x y x yphn bitsao cho( ) ( )2 22 1 2 190 + = x x y y (a) 2 4( 1) 11( 1) 1+ = + + = +xk xxy k x(I).Ta c: 2(2 3) 3 0( )( 1) 1kx k x kIy k x + + = = + (I) c hai nghim phn bit PT 2(2 3) 3 0 ( ) + + = kx k x k bc hai nghim phn bit. 30, .8k k = (2) Khi ta c:1 21 2222mx xmx x+ = +=. Gi ( ) ( )A x x m B x x m1 1 2 2; 2, ; 2 + + . AB2 = 5 2 21 2 1 2( ) 4( ) 5 x x x x + = 21 2 1 2( ) 4 1 x x x x + =m m28 20 0 = mm102 =

= (tho (2)) Vy:m m 10; 2 = = . Cu 78. Cho hm sxyx m1 =+(1). 1) Kho st s bin thin v v th ca hm s (1) khim 1 = . 2) Tm cc gi tr ca tham s m sao cho ng thng (d):y x 2 = +ct th hm s (1) ti hai im A v B sao choAB 2 2 = . Gii - PThonh giao im:x m xxx m x m x m 212( 1) 2 1 0 (* ) = = + + + + + + = d ct th hm s (1) ti hai im A, B phn bit (*) c hai nghim phn bit khcm mm m mx mm m203 2 3 3 2 3 6 3 01 1A >< v > + > = = = (**) Khi gixx1 2,l cc nghim ca (*), ta c x x mx x m1 21 2( 1). 2 1 + = += + Cc giao im ca d v th hm s (1) lAxxBxx1 1 2 2( ; 2), ( ; 2) + + . Suy raAB x x x x xx m m2 2 2 21 2 1 2 1 22( ) 2 ( ) 4 2( 6 3) (= = + = Theo gi thit ta c mm m m mm2 2 12( 6 3) 8 6 7 07 = = = = Kt hp vi iu kin (**) ta cm 7 =l gi tr cn tm. Cu 79. Cho hm s 32xyx+=+ c th (H). Tm m ng thng d :y = 2x + 3m ct (H) ti hai im phn bit sao cho. 4 OA OB = vi O l gc ta . Gii Phng trnh honh giao im ca (H) vi (d) :232 3 2 3(1 ) 6 3 0 (1) ( 2)2xx m x mx m xx+= + + + + = = +

(H) ct (d) ti hai im phn bit A v B (1) c hai nghim phn bit khc -2 GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 36 29 30 33 08 6(1 ) 6 3 0m mmm mA = + > + + = Gi 2 nghim ca pt (1) l 1x ;2x th A(1x ;21x +3m) ; B(2x ;22x +3m) C :. 4 OA OB = 1x2x +(21x +3m)(22x +3m) = - 4 12 15 742 12mm= =Cu 80. Tmtrn(H) : 12xyx +=ccimA,Bsaocho dionthngABbng4vng thng AB vung gc vi ng thng y = x Gii Do AB d : y = x pt AB: y = -x + m Phng trnh honh giao im ca (H) vi ng thng AB :21( ) ( 3) 2 1 0 ( 2) (1)2xx m g x x m x m xx+= + = + + + = = tn ti 2 im A, B th pt (1) cn c hai nghim phn bit;A Bx xv khc 222( )0( 3) 4(2 1) 0( 1) 4 0;4 ( 3).2 2 1 0 (2) 0g x m mm mm m g A >+ + > + > + + + = = Theo Viets ta c : 3. 2 1A BA Bx x mx x m + = += +Mt khc :;A A B By x my x m = + = + 2 2 2 22 2: 4 16 ( ) ( ) 16 ( ) 4 . 81( 3) 4(2 1) 0 2 3 03B A B A B A A BM aAB AB x x y y x x x xmm m m mm= = + = + = = + + = = = +) Vi m = 3 thay vo pt (1) ta c : 26 7 0 3 2 2 x x x y + = = = (3 2; 2); (3 2; 2) (3 2; 2); (3 2; 2) A B hoac A B + + +) Vi m = -1 thay vo (1) ta c : 22 1 0 1 2 2 2 x x x y = = = (1 2; 2 2); (1 2; 2 2) (1 2; 2 2); (1 2; 2 2) A B hoac A B + + + + Kt lun: .... Cu 81. Cho hm s 32xyx+=c th (H). Tm m ng thng d : y = -x + m + 1 ti hai im phn bit A, B sao cho AOB nhn. Gii Phng trnh honh giao im ca (H) vi d :231 ( 2) 2 5 0 ( 2)2xx m x m x m xx+= + + + + + = = phng trnh trn c hai nghim phn bit th : 224 16 00; 22 2( 2) 2 5 0m mx mm m + >A > = + + + = Gi 1 1 2 2( ; 1); ( ; 1) Ax x m Bx x m + + + + l 2 giao im ca (H) v d GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 37 AOB nhn th 2 2 2 2 2 22 1 1 22( ) ( 1) ( 1) AB OA AB x x x m x m < + < + + + + +21 2 1 22 ( 1)( ) ( 1) 0 3 xx m x x m m + + + + < > Kt lun : m > -3 Cu 82. Cho hm s3 2( )2xy Cx+=+ 1) Kho st v v th (C) ca hm s. 2) ng thngy x = ct (C) ti hai im A, B. Tm m ng thngy x m = + ct (C) ti hai im C, D sao cho ABCD l hnh bnh hnh. Gii Honh im A, B l nghim caphng trnh:3 2 1( 1; 1); (2; 2) 3 222x xx A B ABxx+= = =

=+ Ta c: C, D thuc ng thng y = x + m vD AB C nn m= 0 v CD = AB =3 2Phng trnh honh giao im ca d v (C):23 2( 1) 2 2 0 (* )2xx m x m x mx+= + + + =+ d ct (C) ti hai im phn bit phi c: 2910 9 01mm mm >A = + > < Gi C(a;a+m); D(b;b+m) vi a, b l nghim ca phng trnh (*) CD =3 2 22( ) 3 2 a b = 2 0 ( )10 9 910 ( / )m l oaim mm t m = + =

=

KL: m = 10 Cu 83. Cho hm s2 11xyx= (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm m ng thng d:y x m = +ct (C) ti hai im phn bit A, B sao cho AOAB vung ti O. Gii - Phng trnh honh giao im ca (C) v d:x m x m x2( 3) 1 0, 1 + + = =(*) (*) cm m m R22 5 0, A = + > ev (*) khng c nghim x = 1. (*) lun c 2 nghim phn bit l A Bx x , . Theo nh l Vit: A BA Bx x mx x m3. 1 + = = Khi : ( ) ( )A A B BA xx mB xx m ; , ; + +OAB Avung ti O th ( )( )A B A BOA OB x x x m x m . 0 0 = + + + = GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 38 ( ) 2 0 22 = = + + + m m x x m x xB A B A Vy: m = 2. Cu 84. Cho hm s: xyx22+=. 1) Kho st s bin thin v v th (C) ca hm s. 2) Chng minh rngvi mi gi trm th trn (C) lun ccp imA, B nm vhai nhnh ca (C) v tha A AB Bx y mx y m00 + = + = . Gii - Ta c: A A A AB B B Bx y m y x mAB d y x mx y m y x m0, ( ) :0 + = = + e = + + = = + A, B l giao im ca (C) v (d). Phng trnh honh giao im ca (C) v (d): xx m f x x m x m xx22( ) ( 3) (2 2) 0 ( 2)2++ = = + + = = (*). (*) cm m m22 17 0, A = + + > (d) lun ct (C) ti hai im phn bit A, B. VA Bf x x 1. (2) 4 0 2 = < < A> A002/1/

> > 0 3 40 1 2 822m mm m

> s> s1 ;4321;41m mm m 41 s mhoc 21> m Cu 88. Cho hm sy x x3 23 1 = +c th (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm hai im A, B thuc th (C) sao cho tip tuyn ca (C) ti A v B song song vi nhau v di on AB =4 2 . Gii - Gi sAa a a Bb b b3 2 3 2( ; 3 1), ( ; 3 1) + +thuc (C), via b = . V tip tuyn ca (C) ti A v B song song vi nhau nn:y a y b ( ) ( )' '=a a b b a b a b a b a b2 2 2 23 6 3 6 2( ) 0 ( )( 2) 0 = = + =a b b a 2 0 2 + = = . Va b =nna a a 2 1 = =Ta c:AB b a b b a a b a b a b a2 3 2 3 2 2 2 3 3 2 2 2( ) ( 3 1 3 1) ( ) ( 3( )) = + + + = + b a b a abb a b a b a22 3( ) ( ) 3 ( ) 3( )( ) (= + + + b a b a b a ab22 2 2( ) ( ) ( ) 3 3.2 (= + + GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 41 b a b a b a ab22 2 2( ) ( ) ( ) 6 (= + + b a b a ab2 2 2( ) ( ) ( 2 ) = + 2AB b a ab a a a2 2 2 2 2( ) 1 ( 2 ) (2 2 ) 1 ( 2 2) ((= + = + a a a a a22 2 2 4 24( 1) 1 ( 1) 3 4( 1) ( 1) 6( 1) 10 ( ((= + = +( a a a6 4 24( 1) 24( 1) 40( 1) = + MAB 4 2 =nna a a6 4 24( 1) 24( 1) 40( 1) 32 + =a a a6 4 2( 1) 6( 1) 10( 1) 8 0 + = (*) tt a t2( 1) , 0 = > . Khi (*) tr thnh: t t t t t t t3 2 26 10 8 0 ( 4)( 2 2) 0 4 + = + = =a baa b2 3 1( 1) 41 3= = = = = Vy 2 im tho mn YCBT l:A B (3;1), ( 1; 3) . Cu 89. Cho hm sy x x33 = (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn ng thng (d):y x = cc im m t k c ng 2 tip tuyn phn bit vi th (C). Gii -Gi e ( ; ) ( ) Mm m d .PT ng thng A i qua im M v c h s gc kc dng : = ( ) y k x m mA l tip tuyn ca (C) h PT sau c nghim = = 323 ( ) (1)3 3 (2)x x k x m mx k (*). Thay (2) v (1) ta c: + = + = =3 23 2323 (3 3 )( ) 02 3 4 02(* * )3 4x x x x m mx mx mxmx Xt hm s :=322( )3 4xf xx. Tp xc nh : D =\ { 2 3 2 3;3 3} C: f(x) = 4 226 243 4x xx; f(x) = 0 =

= 02xx GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 42 T M k c ng hai tip tuyn ti th hm s (**) c hai nghim phn bit. Da vo bng bin thin ta c: (**) c 2 nghim phn bit =

= 22mm Cc im cn tm l: A(2; 2) vB(2; 2). Cu 90. Cho hm s:2 33+ = x x y . 1) Kho st v v th (C) ca hm s. 2) Tm tt c im trn ng thng y = 4 , sao cho t k c ng 2 tip tuyn ti th (C). Gii Gi M(x0;4) l imcn tm. Phng trnh ng thng (d) qua M vi h s gc k l: y=k(x-xo)+4 (d) l tip tuyn ca th hm s h sau c nghim: 302( ) 4 3 2 (1)3 3 (2)k x x x xk x + = += Thay (2) vo (1) ta c: ( )2 33 2 33 220 0(3 3)( ) 4 3 23 3 3 3 3 22 3 3 2 01 2 (3 2) 3 2 0 (3)oo oo ox x x x xx x x x x x xx x x xx x x x x + = + + = + + = ( + + + + = k c 2 tip tuyn ti th hm s khi v ch khi (3) c ng 2 nghim. 20 02 (3 2) 3 2 0 x x x x + + + = c : +)2 nghim trong 1 nghim bng-1; +)C nghim kp khc -1.Gii tm ng 3 im: (-1;4);(-2/3;4);(2;4)

Cu 91. Cho hm s y x x3 23 2 = + (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn ng thng (d): y = 2 cc im m t k c 3 tip tuyn phn bit vi th (C). Gii -Gie ( ; 2) ( ) Mm d .-2+ +---+- -2 2 330 - 2 33 2+002--+--+ x f(x) f(x) 2 GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 43 PT ng thng A i qua im M v c h s gc kc dng : y kx m ( ) 2 = +A l tip tuyn ca (C) h PT sau c nghim x x kx m x x k3 223 2 ( ) 2 (1)3 6 (2) + = + + = (*). Thay (2) v (1) ta c:x m x mx x x m x3 2 22 3( 1) 6 4 0 ( 2) 2 (3 1) 2 0 ( + + = + = =

= + =22( ) 2 (3 1) 2 0 (3)xf x x m x T M k c 3 tip tuyn n th (C) h (*) c 3 nghim x phn bit(3) c hai nghim phn bit khc 2 A > < > ==50 1 3(2) 02m hoc mfm . Vyt cc im M(m; 2) e (d): y = 2 vi < >=51 32m hoc mm c th k c 3 tip tuyn n (C). Cu 92. Cho hm s y f x mx m x mx3 21( ) ( 1) (4 3 ) 13= = + + +c th l (Cm). 1) Kho st s bin thin v v th ca hm s khi m = 1. 2) Tm cc gi tr m sao cho trn th (Cm) tn ti mt im duy nht c honh m m tip tuyn ti vung gc vi ng thng (d):x y 2 3 0 + = . Gii -(d) c h s gc 12 tip tuyn c h s gck 2 = . Gi x l honh tip im th: f x mx m x m mx m x m2 2'( ) 2 2( 1) (4 3 ) 2 2( 1) 2 3 0 = + + = + + = (1) YCBT (1) c ng mt nghim m. + Num 0 =th (1) x x 2 2 1 = =(loi) + Num 0 = th d thy phng trnh (1) c 2 nghim l mx hayx=m2 31=Do (1) c mt nghim m th mmm m02 3023

Vym hay m203< >. Cu 93. Tm tt c cc gi tr m sao cho trn th( )mC :3 21( 1) (4 3) 13y mx m x m x = + + +tn tinghaiimchonhdngmtiptuyntivunggcvingthng(L): 2 3 0 x y + =Gii GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 44Cch 1: C y = 22( 1) 4 3 mx m x m + + T yu cu bi ton ta c: 1' . 12y| | = |\ .c ng hai nghim dng phn bit 22( 1) 2 3 0 mx m x m + + = c hai nghim phn bit. 2000 1 14 4 1 00' 01 2 200 1 1 2 02 02 32 3003mmmm mm mmm SmmPmmm = == + > = < < A > >< < > < < Vy 1 1 20; ;2 2 3m| | | |e ||\ . \ .l cc gi tr cn tm Cch 2: C2' 2( 1) 4 3 y mx m x m = + + T yu cu bi ton ta c: 1' . 12y| | = |\ .c ng hai nghim dng phn bit 22( 1) 2 3 0 mx m x m + + = (1) c hai nghim phn bit. +) TH1: m = 0 t (1) ta c: x = -1 (loi) +) TH2: 12m = t (1) ta c x = 1(loi)+) Th3: 10;2m m = = T pt (1) c 2 nghim 2 31mx xm= v =iu kin bi ton 2 3 20 03mmm> < (( . Du = xy ra mm31 =

= Vy im M cn tm c ta l: M(3; 3)hocM(1;1) Cu 101.Cho hm s xyx2 32=. 1) Kho st s bin thin v v th(C) ca hm s. 2) Cho Ml im bt k trn (C). Tip tuyn ca (C) ti M ct cc ng tim cn ca (C) ti A vB. Gi I l giao im ca cc ng tim cn. Tm to im M sao cho ng trn ngoi tip tam gic IAB c din tch nh nht. Gii - Gi s xM x xx00 002 3; , 22| | = | |\ ., ( )y xx0201'( )2= Phng trnh tip tuyn (A) vi ( C) ti M: ( )xy x xxx002002 3 1( )22 = + To giao im A, B ca (A) vi hai tim cn l: ( )xA B xx0002 22; ; 2 2; 22| | | |\ . Tathy A BMx x xx x002 2 22 2+ + = = = , A BMy y xyx002 32 2+ = =suyraMltrungimca AB. Mt khc I(2; 2) v AIAB vung ti I nn ng trn ngoi tip tam gic IAB c din tchS = xI M x xxx22 2 2 00 02002 3 1( 2) 2 ( 2) 22( 2)t t t t ( (| | ( ( = + = + > | | ( ( \ . Du = xy ra khi xxxx2002001 1( 2)3( 2) = = = Do im M cn tm l M(1; 1) hoc M(3; 3) GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 49 Cu 102.Cho hm s 11 2+=xxyc th(C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Gi I l giao im ca hai tim cn. Tm im M thuc (C) sao cho tip tuyn ca (C) ti M ct 2 tim cn ti A v B vi chu vi tam gic IAB t gi tr nh nht. Gii - Giao im ca 2 tim cn lI (1; 2) . Gi M||.|

\|+132 ;00xxe (C).+ PTTT ti M c dng:y x xxx02003 3( ) 21( 1)= + + + To cc giao im ca tip tuyn vi 2 tim cn: A||.|

\|+162 ; 10x, B x0(2 1; 2) + Ta c: I ABS I A I B xx001 1 6. 2 1 2.3 62 2 1A= = = =(vdt) + AIAB vung c din tch khng i chu vi AIAB t gi tr nh nht khi IA= IB

= + = =3 13 11 2160000 xxxx Vy c hai im M tha mn iu kin( )M11 3; 2 3 + + , ( )M21 3; 2 3 Khi chu vi AAIB =6 2 3 4 + . Ch : Vi 2 s dng a, b tho ab = S (khng i) th biu thc P =a b a b2 2+ + +nh nht khi v ch khi a = b. Tht vy: P =a b a b2 2+ + + >ab ab ab S 2 2 (2 2) (2 2) + = + = + . Du "=" xy ra a = b. Cu 103.Cho hm s:xyx21+=(C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Cho imA a (0; ) . Tm a t A k c 2 tip tuynti th (C) sao cho 2 tip im tng ng nm v 2 pha ca trc honh. Gii - Phng trnh ng thng d i quaA a (0; )v c h s gc k: y kx a = +d l tip tuyn ca (C) H PT xkx axkx2213( 1) += + = c nghim PT:ax a x a2(1 ) 2( 2) ( 2) 0 + + + = (1)cnghim x 1 = . GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 50 qua A c 2 tip tuyn th (1) phi c 2 nghim phn bitx x1 2, aaaa1123 6 0 A = = ' > = + >(*) Khi ta c: a ax x xxa a1 2 1 22( 2) 2;1 1+ ++ = = vy yx x1 21 23 31 ; 11 1= + = + 2 tip im nm v 2 pha i vi trc honh thy y1 2. 0 < x x1 23 31 . 1 01 1| | | |+ + < || \ . \ . x x x xx x x x1 2 1 21 2 1 2. 2( ) 40. ( ) 1+ + +< + + a 3 2 0 + >a23> Kt hp vi iu kin (*) ta c: aa231 > =. Cu 104.Cho hm sxyx31+=. 1) Kho st s bin thin v v th (C) ca hm s. 2) Cho im o o oM x y ( ; )thuc th (C). Tip tuyn ca (C) ti M0 ct cc tim cn ca (C) ti cc im A v B. Chng minh Mo l trung im ca on thng AB. Gii - o o oM x y ( ; )e (C) yx00411= +.Phng trnh tip tuyn (d) ti M0 :y y x xx0 0204( )( 1) = Giao im ca (d) vi cc tim cn l:A x B y0 0(2 1;1), (1; 2 1) . A B A Bx x y yx y0 0;2 2+ += = M0 l trung im AB. Cu 105.Cho hm s :xyx21+=(C)1) Kho st s bin thin v v th (C) ca hm s. 2) Chng minh rng mi tip tuyn ca th (C) u lp vi hai ng tim cn mt tam gic c din tch khng i. Gii - Gi sMaaa2;1| | + |\ . e (C).PTTT (d) ca (C) ti M: ay y a x aa2( ).( )1+'= + a ay xa a22 23 4 2( 1) ( 1) + = + GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 51 Cc giao im ca (d) vi cc tim cn l: aAa51;1| | + | \ .,B a (2 1;1) . I Aa60;1| |= |\ . I Aa61= ; I B a (2 2; 0)= I B a 2 1 = Din tchI AB A : SI AB A=I A I B1.2= 6 (vdt)PCM. Cu hi tng t i vi hm s xyx2 41=+S: S = 12. Cu 106.Cho hm s y = 12++xx. 1) Kho st s bin thin v v th (C) ca hm s. 2) Gi I l giao im ca 2 ng tim cn,A l mt tip tuyn bt k ca th (C). d l khong cch t I nA. Tm gi tr ln nht ca d. Gii -yx21( 1)' =+. Giao im ca hai ng tim cn l I(1; 1). Gi s xM x Cx0002; ( )1| | +e |+\ . Phng trnh tip tuynA vi thi hm s ti M l: ( )xy x xxx002002 1( )11+ = +++

( ) ( )( )x x y x x x20 0 0 01 1 2 0 + + + + =Khong cch t I n A l d =( )xx0402 11 1++ +=( )( )xx202022111s+ ++

Vy GTLN ca d bng2 khix00 =hoc x02 = .

Cu 107.Cho hm s xyx2 11=. 1) Kho st s bin thin v v th (C) ca hm s. 2) Vit phng trnh tip tuyn ca (C), bit khong cch t im I(1; 2) n tip tuyn bng2 . Gii - Tip tuyn ca (C) ti imMx f x C0 0( ; ( )) ( ) e c phng trnh:y f x x x f x0 0 0'( )( ) ( ) = +x x y x x2 20 0 0( 1) 2 2 1 0 + + = (*)Khong cch t im I(1; 2) n tip tuyn (*) bng2xx0402 221 ( 1) =+ xx0002 =

= Cc tip tuyn cn tm :x y 1 0 + = v x y 5 0 + = GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 52Cu 108.Cho hm s xyx11+=(C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn Oy tt c cc im t k c duy nht mt tip tuyn ti (C). Gii - Gi oM y (0; )l im cn tm. PT ng thng qua M c dng: oy kx y = +(d) (d) l tip tuyn ca (C) o o o oxkx y y x y x y xxkkxx2221( 1) 2( 1) 1 0 (1)1221;( 1)( 1) += + + + + = = == (*) YCBT h (*) c 1nghim(1) c 1 nghim khc 1 oooo o ooyyx y kx y y yx y k21 11; 1 81 2' ( 1) ( 1)( 1) 00; 1 22A= = = = =

v = = + + = = = = Vy c 2 im cn tm l: M(0; 1) v M(0; 1). Cu 109.Cho hm s xyx2 11+=+. 1) Kho st s bin thin v v th (C) ca hm s. 2) Vit phng trnh tip tuyn ca th (C), bit rng tip tuyn cch u hai imA(2; 4),B(4; 2). Gii - Gi x0 l honh tip im ( x01 = ).PTTT (d) l xy x xxx002002 1 1( )1( 1)+= +++ x x y x x2 20 0 0( 1) 2 2 1 0 + + + + =Ta c:dAd dBd ( , ) ( , ) =x x x x x x2 2 2 20 0 0 0 0 02 4( 1) 2 2 1 4 2( 1) 2 2 1 + + + + = + + + + +x x x0 0 01 0 2 = v = v = Vy c ba phng trnh tip tuyn:y xy xy x1 5; 1; 54 4= + = + = +Cu 110.Cho hm s 2 11=xyx. 1) Kho st s bin thin v v th (C) ca hm s. 2) Gi I l giao im ca hai ng tim cn, A l im trn (C) c honh l a. Tip tuyn ti A ca (C) ct hai ng tim cn ti P v Q. Chng t rng A l trung im ca PQ v tnh din tch tam gic IPQ. Gii - aI A aa2 1(1; 2), ;1| | | \ .. PT tip tuyn d ti A: 21 2 1( )(1 ) 1= + ay x aa a GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 53 Giao im ca tim cn ng v tip tuyn d:21;1| | | \ .aPa Giao im ca tim cn ngang v tip tuyn d: Q a (2 1; 2) Ta c:P Q Ax x a x 2 2 + = = . VyA l trung im ca PQ. IP = 2 221 1aa a+ = ; IQ =2( 1) a SIPQ = 12IP.IQ = 2 (vdt) Cu 111.Cho hm s xyx2 32=(C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Vit phng trnh tip tuyn ti im M thuc (C) bit tip tuyn ct tim cn ng v tim cn ngang ln lt ti A, B sao cho csin gc ABIbng 417, vi I l giao 2 tim cn. Gii - I(2; 2). Gi xM x Cx0002 3; ( )2| | e |\ .,x02 =Phng trnh tip tuyn Ati M:xy x xxx002002 3 1( )2( 2)= +

Giao im ca A vi cc tim cn:xAx002 22;2| | |\ .,B x0(2 2; 2) . Do ABI4cos17=nn I AABII B1tan4= =I B I A2 216. = x40( 2) 16 =xx0004 =

= Kt lun:TiM30;2| | |\ . phng trnh tip tuyn: y x1 34 2= +Ti M54;3| | |\ . phng trnh tip tuyn: y x1 74 2= + Cu 112.Cho hm s 12 1xyx+= 1) Kho st s bin thin v v th ca hm s. 2) Tm gi tr nh nht ca m sao cho tn ti t nht mt im M e(C) m tip tuyn ti M ca (C) to vi hai trc ta mt tam gic c trng tm nm trn ng thng2 1 y m = Gii Gi( ; )o oM x y l ta tip im Phng trnh tip tuyn ti im( ; )o oM x y l 23( )(2 1)o ooy x x yx= + GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 54Gi A, B l giao im ca tip tuyn vi trc honh, trc tung tng ng Nn ta c: 222 4 1(2 1)o oBox xyx+ =v trng tm G ca tam gic OAB c 222 4 13(2 1)o oGox xyx+ = Theo gi thit trng tm nm trn ng thng y = 2m + 1 nn 222 4 12 13(2 1)o oox xmx+ = Ta li c: 2 2 2 22 22 4 1 6 (2 1) 61 1(2 1)3(2 1) (2 1)o o o o ooo ox x x x xxx x+ = = > Vy, tn ti t nht mt im M tha mn iu kin bi ton 2m - 1> 13 13m > GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 55 PHN 5: BIN LUN S NGHIM CA PHNG TRNH Cu 113.Cho hm sy x x3 23 1 = + + . 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm m phng trnhx x m m3 2 3 23 3 = c ba nghim phn bit. - PTx x m m3 2 3 23 3 = x x m m3 2 3 23 1 3 1 + + = + + . tk m m3 23 1 = + +S nghim ca PT bng s giao im ca th (C) vi ng thng d:y k =Da vo th (C) ta c PT c 3 nghim phn bit k 1 5 < 0) -x x m4 222 1 l og 0 + + =x x m4 222 1 l og + = (*) + S nghim ca (*)l s giao im ca 2 thy x x4 22 1 = +vy m2l og = + T th suy ra: m102< < m12= m112 < < m 1 = m 1 >2 nghim 3 nghim4 nghim 2 nghim v nghim Cu 116.Cho hm sy f x x x4 2( ) 8 9 1 = = + . 1) Kho st s bin thin v v th (C) ca hm s. 2) Da vo th (C) hy bin lun theo m s nghim ca phng trnh: x x m4 28cos 9cos 0 + =vix [ 0; ] t e- Xt phng trnh:x x m4 28cos 9cos 0 + =vix [ 0; ] t e (1) tt x cos = , phng trnh (1) tr thnh:t t m4 28 9 0 + =(2) Vx [ 0; ] t e nnt [ 1;1] e ,giaxvtcstngngmtimt,dosnghimca phng trnh (1) v (2) bng nhau. Ta c:t t m4 2(2) 8 9 1 1 + = (3) GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 56Gi(C1):y t t4 28 9 1 = + vit [ 1;1] e v(d):y m 1 = .Phngtrnh(3)lphngtrnh honh giao im ca (C1) v (d).Ch rng (C1) ging nh th (C) trong minx 1 1 s s . Da vo th ta c kt lun sau: m 0 < m 0 = m 0 1 < < m81132s < m8132= m8132>v nghim1 nghim2 nghim4 nghim2 nghimv nghim Cu 117.Cho hm s xyx3 42= (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm cc gi tr ca m phng trnh sau c 2 nghim trn on 20;3t( ( : x x m x x6 6 4 4si n cos( si n cos ) + = +- Xt phng trnh: x x m x x6 6 4 4si n cos( si n cos ) + = + (*) x m x2 23 11 si n 2 1 si n 24 2| | = |\ . x m x2 24 3si n 2 2 (2 si n 2 ) = (1) tt x2si n 2 = . Vix20;3t(e ( th| | t 0;1 e . Khi (1) tr thnh: tmt3 422= vit 0;1( e Nhn xt : vi mit 0;1( e ta c : x tx tx tsi n2si n2si n2

= =

= (*) c 2 nghim thuc on 20;3t( ( tht t3 3;1 ;12 4 | |e e | | | . . Da vo th (C) ta c: y m y m3 7(1) 2 1 24 5| |< s < s |\ . m1 72 10< s . Cu 118.Cho hm s 1.1xyx+= 1) Kho st s bin thin v v th (C) ca hm s. 2) Bin lun theo m s nghim ca phng trnh 1.1xmx+= - S nghim ca 11xmx+= bng s giao im ca th (C'): 11xyx+= v. y m =Da vo th ta suy ra c: 1; 1 < > m m 1 = m 1 1 < s m2 nghim1 nghimv nghim GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 57 PHN 6: IM C BIT CA TH Cu 119.Cho hm s 33 2 y x x = + +(C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm 2 im trn th hm s sao cho chng i xng nhau qua tm M(1; 3). Gii - Gi ( )A x y0 0; ,Bl im i xng vi A qua imM( 1; 3) ( )B x y0 02 ; 6 AB C , ( ) e y x xy x x30 0 030 0 03 26 ( 2 ) 3( 2 ) 2= + + = + + ( ) ( )x x x x x x33 20 0 0 0 0 06 3 2 2 3 2 2 6 12 6 0 = + + + + + + =x y0 01 0 = =Vy 2 im cn tm l: ( )1; 0 v ( )1; 6 Cu 120.Cho hm s33 2 y x x = + +(C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn (C) hai im i xng nhau qua ng thng d:x y 2 2 0 + = . Gii - Gi( ) ( )1 1 2 2; ; ; M x y N x ythuc (C) l hai im i xng qua ng thng dI l trung im ca AB nn 1 2 1 2;2 2x x y yI+ + | | |\ ., ta cI d eC: ( ) ( )3 31 1 2 21 2 1 23 2 3 22. 22 2 2x x x xy y x x + + + + ++ += = +

( ) ( ) ( ) ( )3 1 21 2 1 2 1 2 1 2 1 22 21 1 2 203 3 21+ = + + + + + = + + =x xx x x x x x x x x xx x x x Li c:( ) ( )2 1 2 1.1 .2 0 MN d x x y y + = ( ) ( )( )2 2 2 22 1 2 1 1 1 2 2 1 1 2 277 2 02 + + = + + = x x x x x x x x x x x x- Xt 1 20 x x + =1 27 7;2 2x x = = - Xt 2 2 2 21 2 1 1 2 22 21 1 2 21 29147524x x x x x xx x x xx x + = + = + + = = v nghim Vy 2 im cn tm l: 7 1 7 7 1 7; 2 ; ; 22 2 2 2 2 2| | | | + || ||\ . \ . GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 58Cu 121.Cho hm sxy x x321133 3= + + . 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn th (C) hai im phn bit M, N i xng nhau qua trc tung. Gii - Hai imMx y Nx y C1 1 2 2( ; ), ( ; ) ( ) ei xng nhau qua Oy x xy y2 11 20 = == x xx xx x x x22 13 32 3 1 21 1 2011 113 33 3 3 3 = = + + = + + xx1233 == hoc xx1233 = =

Vy hai im thuc th (C) v i xng qua Oy l:M N16 163; , 3;3 3| | | | ||\ . \ .. Cu 122.Cho hm s xyx2 11=+ (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm im M thuc th (C) tip tuyn ca (C) ti Mvi ng thng i quaM v giao im hai ng tim cn c tch cc h s gc bng 9. Gii - Giao im 2 tim cn lI ( 1; 2) .Gi M II MM Iy yM x C kx x xx02003 3; 2 ( )1( 1) | | e = = |+ +\ . + H s gc ca tip tuyn ti M: ( )Mk y xx0203( )1'= =+ + YCBT M I Mk k . 9 = xx000 2 =

= . Vy c 2 im M tha mn: M(0; 3) v M(2; 5) Cu 123.Cho hm s 2 11xyx+=+(C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn (C) nhng im c tng khong cch n hai tim cn ca (C) nh nht. Gii - GiMx y0 0( ; ) e (C), ( x01 = ) th xyx x000 02 1 121 1+= = + + Gi A, B ln lt l hnh chiu ca M trn TC v TCN th: MA x MB yx0 0011 , 21= + = =+ GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 59 p dng BT C-si ta c:MA MB MA MB xx0012 . 2 1 . 21+ > = + =+ MA + MB nh nht bng 2 khi xxxx00000 1121 =+ = = +.Vy ta c hai im cn tm l (0; 1) v (2; 3). Cu hi tng t: a) 2 11=+xyxS: 01 3 x = Cu 124.Cho hm s xyx3 42=(C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm cc im thuc (C) cch u 2 tim cn. Gii - GiMxy ( ; ) e (C) v cch u 2 tim cn x = 2 v y = 3. Ta c: x xx y x xx x3 42 3 2 2 22 2 = = = xxxxx1( 2)42 = = = Vy c 2 im tho mn bi l : M1( 1; 1) v M2(4; 6) Cu 125.Cho hm s xyx2 41=+. 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn (C) hai im i xng nhau qua ng thng MN bit M(3; 0) v N(1; 1). Gii -MN (2; 1) = Phng trnh MN:x y 2 3 0 + + = . Phng trnh ng thng(d) MN c dng:y x m 2 = + . Phng trnh honh giao im ca (C) v (d): xx mx2 421= ++ x mx m x22 4 0( 1) + + + = = (1) (d) ct (C) ti hai im phn bit A, B m m2 8 32 0 A = > (2) Khi Ax x m Bx x m1 1 2 2( ; 2 ), ( ; 2 ) + +vix x1 2,l cc nghim ca (1) Trung im ca AB l x xI x x m1 21 2;2| | ++ + |\ .m mI ;4 2| | |\ . (theo nh l Vi-et) A, B i xng nhau qua MN I e MN m 4 = Suy ra (1) xx xx2 02 4 02 = = = A(0; 4), B(2; 0). GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 60Cu 126.Cho hm s 21xyx=. 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn th (C) hai im B, C thuc hai nhnh sao cho tam gic ABC vung cn ti nh A vi A(2; 0). Gii - Ta cC yx2( ) : 21= +. GiB b C cb c2 2; 2 , ; 21 1+ + | | | | ||\ . \ .vib c 1 + ++xx 6 s d . Khong cch d ln nht bng6khi ( ) 3 1 3 1 ) 1 () 1 (902020 20 = = + + =+x x xx. Vy c hai im cn tm l:( ) 3 2 ; 3 1 + M hoc( ) 3 2 ; 3 1 + M H KBAC GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 61 Cu 128.Cho hm s xyx22 1+=. 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm nhng im trn th (C) cch u hai im A(2; 0) v B(0; 2). Gii - PT ng trung trc an AB:y x = . Nhng im thuc th cch u A v B c honh l nghim ca PT: xxx22 1+= xx xx21 521 01 52

= = + =

Hai im cn tm l: 1 5 1 5 1 5 1 5, ; ,2 2 2 2| | | | + + || ||\ . \ . Cu 129.Cho hm s=+31xyx . 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm trn hai nhnh ca th (C) hai im A v B sao cho AB ngn nht. Gii - Tp xc nh D =R \ { 1} . Tim cn ngx 1 = . Gi sA a B ba b4 41 ;1 , 1 ;1| | | | + + ||\ . \ . (via b 0, 0 > > ) l 2 im thuc 2 nhnh ca (C) AB a b a b ab aba b abab ab22 2 22 2 2 21 1 16 16 64( ) 16 ( ) 1 4 1 4 32| |((= + + + = + + > + = + > | ((\ . AB nh nht = = = = = = =444 2 4 164 4a ba bAB a bab aab Khi :( ) ( ) A B4 4 4 41 4;1 64 , 1 4;1 64 + + . Cu 130.Cho hm s 4 22 1 y x x = +1) Kho st s bin thin v v th (C) ca hm s. 2)TmtahaiimP.Qthuc(C)saochongthngPQsongsongvitrchonhv khong cch t im cc i ca (C) n ng thng PQ bng 8 Gii Phng trnh ng thng PQ c dng: y = m (m > 0) V im cc i A(0;1) cch PQ mt khong bng 8 nn m = 9 Khi , honh P, Q l nghim ca phng trnh: 4 22 8 0 2 x x x = = Vy, P(-2;9), Q(2;9) hoc P(2;9); Q(-2;9) GV: Lu Huy Thng0913.283.238 GIO DC HNG PHC - NI KHI U C M 62Cu 131.Cho hm s 3 21 533 3y x x x = + + 1) Kho st s bin thin v v th hm s. 2) GiAvBlgiaoimca(C)vtrcOx.Chngminhrng,trnthtntihaiim cng nhn on AB di mt gc vung. Gii Phng trnh honh giao im ca th vi trc honh:3 21 5 13 053 3xx x xx =+ + = = A(-5;0) v B(1;0) Gi M thuc (C) 3 21 5; 33 3M a a a a| |+ + |\ .khc A, B 3 2 3 21 5 1 55; 3 ; 1; 33 3 3 3AM a a a a BM a a a a| | | |+ + + + + ||\ . \ . Theo gi thit:. 0 AM BM AMBM = 2 21( 5)( 1) ( 1) ( 5) 03a a a a (+ + + = ( Do M khc A, B nn a khc -5 v a khc 1 nn phng trnh tng ng: 3 4 3 211 ( 1) ( 5) 0 2 12 14 4 0 (* )9a a hay a a a a + + = + + + =t: 4 3 22 12 14 4 y a a a a = + + + c tp xc nh l3 2' 4 6 12 14; ' 0 y a a a y = + + = c 1 nghim thc 7 20432 16o oa y~ ~ T bng bin thin ta c: (*) lun c hai nghim khc 1 v -5 Vy lun tn ti 2 im cng nhn on AB di mt gc vung + +-+--+0oy