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Gia s Thnh c www.daythem.com.vn KSHS 01: TNH N IU CA HM S Cho hm s y (m 1) x mx (3m 2) x (1) 1) Kho sỏt s bin thiờn v v th (C) ca hm s (1) m 2) Tỡm tt c cỏc giỏ tr ca tham s m hm s (1) ng bin trờn xỏc nh ca nú Cõu Tp xỏc nh: D = R y (m 1)x 2mx 3m (1) ng bin trờn R y 0, x m Cho hm s y x 3x mx (1) 1) Kho sỏt s bin thiờn v v th ca hm s (1) m 2) Tỡm tt c cỏc giỏ tr ca tham s m hm s (1) ng bin trờn khong (;0) Cõu m Cho hm s y x3 3(2m 1)x 6m(m 1) x cú th (Cm) 1) Kho sỏt s bin thiờn v v th ca hm s m = 2) Tỡm m hm s ng bin trờn khong (2; ) Cõu y ' x 6(2m 1)x 6m(m 1) cú (2m 1)2 4(m2 m) x m Hm s ng bin trờn cỏc khong (; m), (m 1; ) y' x m Do ú: hm s ng bin trờn (2; ) m m Cho hm s y x3 (1 2m) x2 (2 m) x m 1) Kho sỏt s bin thiờn v v th (C) ca hm s m = 2) Tỡm m hm ng bin trờn 0; Cõu Hm ng bin trờn (0; ) y 3x 2(1 2m)x (2 m) vi x (0; ) 3x x f (x) m vi x (0; ) 4x 2(6 x x 3) 73 6x2 x x Ta cú: f ( x ) 12 (4 x 1) Lp bng bin thiờn ca hm f ( x ) trờn (0; ) , t ú ta i n kt lun: 73 73 f m m 12 Cho hm s y x4 2mx2 3m (1), (m l tham s) 1) Kho sỏt s bin thiờn v v th ca hm s (1) m = 2) Tỡm m hm s (1) ng bin trờn khong (1; 2) Ta cú y ' x3 4mx x( x m) Cõu + m , y 0, x m tho + m , y cú nghim phõn bit: m , 0, Hm s (1) ng bin trờn (1; 2) ch WWW.ToanCaBa.Net m m m Vy m ;1 Trang mx (1) xm 1) Kho sỏt s bin thiờn v v th ca hm s (1) m 2) Tỡm tt c cỏc giỏ tr ca tham s m hm s (1) nghch bin trờn khong (;1) Cõu Cho hm s y Tp xỏc nh: D = R \ {m} y m2 ( x m)2 Hm s nghch bin trờn tng khong xỏc nh y m hm s (1) nghch bin trờn khong (;1) thỡ ta phi cú m m Kt hp (1) v (2) ta c: m (1) (2) KSHS 02: CC TR CA HM S Cho hm s y x 3x mx m (m l tham s) cú th l (Cm) 1) Kho sỏt s bin thiờn v v th hm s m = 2) Xỏc nh m (Cm) cú cỏc im cc i v cc tiu nm v hai phớa i vi trc honh PT honh giao im ca (C) v trc honh: x x3 3x mx m (1) (2) g( x ) x x m (Cm) cú im cc tr nm v phớa i vi trc 0x PT (1) cú nghim phõn bit (2) cú nghim phõn bit khỏc m m3 g(1) m Cõu Cho hm s y x (2m 1) x (m2 3m 2) x (m l tham s) cú th l (Cm) 1) Kho sỏt s bin thiờn v v th hm s m = 2) Xỏc nh m (Cm) cú cỏc im cc i v cc tiu nm v hai phớa ca trc tung Cõu y 3x 2(2m 1) x (m2 3m 2) (Cm) cú cỏc im C v CT nm v hai phớa ca trc tung PT y cú nghim trỏi du 3(m2 3m 2) m Cho hm s y x3 mx (2m 1) x (m l tham s) cú th l (Cm) 1) Kho sỏt s bin thiờn v v th hm s m = 2) Xỏc nh m (Cm) cú cỏc im cc i, cc tiu nm v cựng mt phớa i vi trc tung Cõu TX: D = R ; y x 2mx 2m th (Cm) cú im C, CT nm cựng phớa i vi trc tung y cú nghim phõn m m2 2m bit cựng du 2m m Cõu 10 Cho hm s y x3 3x mx (m l tham s) cú th l (Cm) 1) Kho sỏt s bin thiờn v v th hm s m = 2) Xỏc nh m (Cm) cú cỏc im cc i v cc tiu cỏch u ng thng y x Trang Gia s Thnh c www.daythem.com.vn Ta cú: y ' 3x x m Hm s cú C, CT y ' 3x2 x m cú nghim phõn bit x1 ; x2 ' 3m m (*) Gi hai im cc tr l A x1; y1 ; B x2 ; y2 m 2m Thc hin phộp chia y cho y ta c: y x y ' x 3 m m 2m 2m y1 y x1 x1 ; y2 y x2 x2 3 m 2m Phng trỡnh ng thng i qua im cc tr l : y x Cỏc im cc tr cỏch u ng thng y x xy trng hp: TH1: ng thng i qua im cc tr song song hoc trựng vi ng thng y x 2m m (tha món) TH2: Trung im I ca AB nm trờn ng thng y x y y x x m 2m y I xI x1 x2 x1 x2 2 2m 2m m0 Vy cỏc giỏ tr cn tỡm ca m l: m 0; Cõu 11 Cho hm s y x 3mx 4m3 (m l tham s) cú th l (Cm) 1) Kho sỏt s bin thiờn v v th hm s m = 2) Xỏc nh m (Cm) cú cỏc im cc i v cc tiu i xng qua ng thng y = x Ta cú: y 3x 6mx ; y x hm s cú cc i v cc tiu thỡ m x 2m uur th hm s cú hai im cc tr l: A(0; 4m3), B(2m; 0) AB (2m; 4m3 ) Trung im ca on AB l I(m; 2m3) AB d A, B i xng qua ng thng d: y = x 2m3 4m m I d 2m m Cõu 12 Cho hm s y x 3mx 3m 1) Kho sỏt s bin thiờn v v th ca hm s m = 2) Vi giỏ tr no ca m thỡ th hm s cú im cc i v im cc tiu i xng vi qua ng thng d: x 8y 74 y 3x 6mx ; y x x 2m Hm s cú C, CT PT y cú nghim phõn bit m Khi ú im cc tr l: A(0; 3m 1), B(2m;4m3 3m 1) AB(2m;4m3 ) Trung im I ca AB cú to : I (m;2m3 3m 1) ng thng d: x 8y 74 cú mt VTCP u (8; 1) WWW.ToanCaBa.Net Trang 3 I d m 8(2m 3m 1) 74 A v B i xng vi qua d m2 AB u AB d Cõu 13 Cho hm s y x 3x mx (1) 1) Kho sỏt s bin thiờn v v th ca hm s m = 2) Vi giỏ tr no ca m thỡ th hm s (1) cú cỏc im cc i v im cc tiu i xng vi qua ng thng d: x 2y Ta cú y x3 3x mx y ' 3x x m Hm s cú cc i, cc tiu y cú hai nghim phõn bit 3m m 1 Ta cú: y x y m x m 3 3 Ti cỏc im cc tr thỡ y , ú ta cỏc im cc tr tha phng trỡnh: y m x m 3 Nh vy ng thng i qua cỏc im cc tr cú phng trỡnh y m x m 3 nờn cú h s gúc k1 m d: x 2y y x d cú h s gúc k2 2 hai im cc tr i xng qua d thỡ ta phi cú d 12 k1k2 m m 23 Vi m = thỡ th cú hai im cc tr l (0; 0) v (2; 4), nờn trung im ca chỳng l I(1; 2) Ta thy I d, ú hai im cc tr i xng vi qua d Vy: m = Cõu 14 Cho hm s y x 3(m 1) x x m (1) cú th l (Cm) 1) Kho sỏt s bin thiờn v v th ca hm s m = 2) Vi giỏ tr no ca m thỡ th hm s cú im cc i v im cc tiu i xng vi qua ng thng d: y x y ' 3x 6(m 1) x Hm s cú C, CT ' 9(m 1)2 3.9 m (; 3) (1 3; ) m Ta cú y x y 2(m 2m 2) x 4m 3 Gi s cỏc im cc i v cc tiu l A( x1; y1 ), B( x2 ; y2 ) , I l trung im ca AB y1 2(m2 2m 2) x1 4m ; y2 2(m2 2m 2) x2 4m x x 2(m 1) v: x1.x2 Vy ng thng i qua hai im cc i v cc tiu l y 2(m2 2m 2)x 4m Trang Gia s Thnh c A, B i xng qua (d): y www.daythem.com.vn AB d m x I d Cõu 15 Cho hm s y x 3(m 1) x x m , vi m l tham s thc 1) Kho sỏt s bin thiờn v v th ca hm s ó cho ng vi m 2) Xỏc nh m hm s ó cho t cc tr ti x1 , x2 cho x1 x2 Ta cú y' 3x 6(m 1) x + Hm s t cc i, cc tiu ti x1 , x2 PT y' cú hai nghim phõn bit x1 , x2 PT x 2(m 1) x cú hai nghim phõn bit l x1 , x2 m ' (m 1) (1) m + Theo nh lý Viet ta cú x1 x2 2(m 1); x1 x2 Khi ú: x1 x2 x1 x2 x1 x2 4m 12 12 (m 1)2 m (2) + T (1) v (2) suy giỏ tr ca m cn tỡm l m v m Cõu 16 Cho hm s y x (1 2m) x (2 m) x m , vi m l tham s thc 1) Kho sỏt s bin thiờn v v th ca hm s ó cho ng vi m 2) Xỏc nh m hm s ó cho t cc tr ti x1, x2 cho x1 x2 Ta cú: y ' 3x 2(1 2m)x (2 m) Hm s cú C, CT y ' cú nghim phõn bit x1, x2 (gi s x1 x2 ) m ' (1 2m) 3(2 m) 4m m m 2 (*) 2(1 2m) x1 x2 Hm s t cc tr ti cỏc im x1, x2 Khi ú ta cú: m x x 2 1 x1 x2 x1 x2 x1 x2 x1x2 29 29 4(1 2m)2 4(2 m) 16m2 12m m m 8 Kt hp (*), ta suy m 29 m x (m 1) x 3(m 2) x , vi m l tham s thc 3 1) Kho sỏt s bin thiờn v v th ca hm s ó cho ng vi m 2) Xỏc nh m hm s ó cho t cc tr ti x1, x2 cho x1 x2 Cõu 17 Cho hm s y Ta cú: y x 2(m 1) x 3(m 2) WWW.ToanCaBa.Net Trang Hm s cú cc i v cc tiu y cú hai nghim phõn bit x1, x2 m2 5m (luụn ỳng vi m) x 2m x x 2(m 1) Khi ú ta cú: x1x2 3(m 2) x2 x2 3(m 2) 8m2 16m m 34 Cõu 18 Cho hm s y x mx 3x 1) Kho sỏt s bin thiờn v v th ca hm s m = 2) Tỡm m hm s cú hai im cc tr x1, x2 tha x1 x2 y 12 x 2mx Ta cú: m2 36 0, m hm s luụn cú cc tr x1, x2 x1 x2 m Khi ú: x1 x2 x1 x2 Cõu hi tng t: m a) y x 3x mx ; x1 2x2 S: m 105 Cõu 19 Cho hm s y (m 2)x 3x mx , m l tham s 1) Kho sỏt s bin thiờn v v th (C) ca hm s m = 2) Tỡm cỏc giỏ tr ca m cỏc im cc i, cc tiu ca th hm s ó cho cú honh l cỏc s dng Cỏc im cc i, cc tiu ca th hm s ó cho cú honh l cỏc s dng PT y ' 3(m 2) x x m = cú nghim dng phõn bit a (m 2) ' 3m(m 2) ' m 2m m m P m m m 3(m 2) m m S m2 Cõu 20 Cho hm s y x 3x (1) 1) Kho sỏt s bin thiờn v v th ca hm s (1) 2) Tỡm im M thuc ng thng d: y 3x tng khong cỏch t M ti hai im cc tr nh nht Cỏc im cc tr l: A(0; 2), B(2; 2) Xột biu thc g( x, y) 3x y ta cú: g( x A , yA ) 3x A y A 0; g( xB , yB ) 3xB yB im cc i v cc tiu nm v hai phớa ca ng thng d: y 3x Do ú MA + MB nh nht im A, M, B thng hng M l giao im ca d v AB Phng trỡnh ng thng AB: y x Trang Gia s Thnh c www.daythem.com.vn x y 3x Ta im M l nghim ca h: M ; 5 y x y Cõu 21 Cho hm s y x (12m) x (2 m) x m (m l tham s) (1) 1) Kho sỏt s bin thiờn v v th hm s (1) m = 2) Tỡm cỏc giỏ tr ca m th hm s (1) cú im cc i, im cc tiu, ng thi honh ca im cc tiu nh hn y 3x 2(1 2m) x m g( x) YCBT phng trỡnh y cú hai nghim phõn bit x1, x2 tha món: x1 x2 4m2 m g(1) 5m m S 2m y x3 3mx2 3(m2 1) x m3 m (1) 1) Kho sỏt s bin thiờn v v th ca hm s (1) m = 2) Tỡm m hm s (1) cú cc tr ng thi khong cỏch t im cc i ca th hm s n gc ta O bng ln khong cỏch t im cc tiu ca th hm s n gc ta O Cõu 22 Cho hm s Ta cú y 3x 6mx 3(m2 1) Hm s (1) cú cc tr thỡ PT y cú nghim phõn bit x2 2mx m2 cú nhim phõn bit 0, m Khi ú: im cc i A(m 1;2 2m) v im cc tiu B(m 1; 2m) m 2 Ta cú OA 2OB m2 6m m 2 Cõu 23 Cho hm s y x 3mx 3(1 m2 ) x m3 m2 (1) 1) Kho sỏt s bin thiờn v v th ca hm s (1) m 2) Vit phng trỡnh ng thng qua hai im cc tr ca th hm s (1) y 3x 6mx 3(1 m2 ) PT y cú 0, m th hm s (1) luụn cú im cc tr ( x1; y1 ), ( x2 ; y2 ) Chia y cho y ta c: Khi ú: m y x y x m m 3 y1 x1 m2 m ; y2 x2 m2 m PT ng thng qua hai im cc tr ca th hm s (1) l y x m2 m Cõu 24 Cho hm s y x3 3x mx cú th l (Cm) 1) Kho sỏt s bin thiờn v v th ca hm s m = 2) Tỡm m (Cm) cú cỏc im cc i, cc tiu v ng thng i qua cỏc im cc tr song song vi ng thng d: y x WWW.ToanCaBa.Net Trang Ta cú: y ' 3x x m Hm s cú C, CT y ' 3x2 x m cú nghim phõn bit x1 ; x2 ' 3m m (*) Gi hai im cc tr l A x1; y1 ; B x2 ; y2 m 2m Thc hin phộp chia y cho y ta c: y x y ' x 3 m m 2m 2m y1 y x1 x1 ; y2 y x2 x2 3 m 2m Phng trỡnh ng thng i qua im cc tr l d: y x ng thng i qua cỏc im cc tr song song vi d: y x 2m m (tha món) m Cõu 25 Cho hm s y x3 3x mx cú th l (Cm) 1) Kho sỏt s bin thiờn v v th ca hm s m = 2) Tỡm m (Cm) cú cỏc im cc i, cc tiu v ng thng i qua cỏc im cc tr to vi ng thng d: x 4y mt gúc 450 Ta cú: y ' 3x x m Hm s cú C, CT y ' 3x2 x m cú nghim phõn bit x1 ; x2 ' 3m m (*) Gi hai im cc tr l A x1; y1 ; B x2 ; y2 m 2m Thc hin phộp chia y cho y ta c: y x y ' x 3 m m 2m 2m y1 y x1 x1 ; y2 y x2 x2 3 m 2m Phng trỡnh ng thng i qua im cc tr l : y x 2m t k ng thng d: x 4y cú h s gúc bng 39 1 k m k k k 10 4 Ta cú: tan 45 1 k m k k k 4 Kt hp iu kin (*), suy giỏ tr m cn tỡm l: m Cõu 26 Cho hm s y x 3x m (1) 1) Kho sỏt s bin thiờn v v th ca hm s (1) m ã 2) Xỏc nh m th ca hm s (1) cú hai im cc tr A, B cho AOB 1200 Trang Gia s Thnh c www.daythem.com.vn Ta cú: y 3x x ; y x y m x y m Vy hm s cú hai im cc tr A(0 ; m) v B(2 ; m + 4) uur uur ã OA (0; m), OB (2; m 4) AOB 1200 thỡ cos AOB m m(m 4) m2 (m 4)2 2m(m 4) 2 3m 24m 44 m2 (m 4)2 m 12 12 m m Cõu 27 Cho hm s y x 3mx 3(m2 1) x m3 (Cm) 1) Kho sỏt s bin thiờn v v th ca hm s (1) m 2) Chng minh rng (Cm) luụn cú im cc i v im cc tiu ln lt chy trờn mi ng thng c nh y 3x 6mx 3(m2 1) ; y x m x m x t im cc i M(m 1;2 3m) chy trờn ng thng c nh: y 3t x t im cc tiu N (m 1; m) chy trờn ng thng c nh: y 3t (1) x mx 2 1) Kho sỏt s bin thiờn v v th ca hm s (1) m 2) Xỏc nh m th ca hm s (1) cú cc tiu m khụng cú cc i Cõu 28 Cho hm s y x y x3 2mx x( x m) y x m th ca hm s (1) cú cc tiu m khụng cú cc i PT y cú nghim m Cõu 29 Cho hm s y f ( x) x 2(m 2) x m2 5m (Cm ) 1) Kho sỏt s bin thiờn v v th (C) hm s m = 2) Tỡm cỏc giỏ tr ca m th (Cm ) ca hm s cú cỏc im cc i, cc tiu to thnh tam giỏc vuụng cõn x Ta cú f x x3 4(m 2) x x m Hm s cú C, CT PT f ( x ) cú nghim phõn bit m (*) Khi ú to cỏc im cc tr l: A 0; m2 5m , B m;1 m , C m;1 m uur uuur AB m; m2 4m , AC m; m2 4m Do ABC luụn cõn ti A, nờn bi toỏn tho ABC vuụng ti A AB AC m 23 m (tho (*)) WWW.ToanCaBa.Net Trang Cm Cõu 30 Cho hm s y x 2(m 2) x m 5m 1) Kho sỏt s bin thiờn v v th hm s m = 2) Vi nhng giỏ tr no ca m thỡ th (Cm) cú im cc i v im cc tiu, ng thi cỏc im cc i v im cc tiu lp thnh mt tam giỏc u x Ta cú f x x3 4(m 2) x x m Hm s cú C, CT PT f ( x ) cú nghim phõn bit m (*) Khi ú to cỏc im cc tr l: A 0; m2 5m , B m;1 m , C m;1 m uur uuur AB m; m2 4m , AC m; m2 4m Do ABC luụn cõn ti A, nờn bi toỏn tho A 600 cos A AB AC AB AC m 23 Cõu hi tng t i vi hm s: y x 4(m 1)x 2m Cõu 31 Cho hm s y x 2mx m2 m cú th (Cm) 1) Kho sỏt s bin thiờn v v th hm s m = 2) Vi nhng giỏ tr no ca m thỡ th (Cm) cú ba im cc tr, ng thi ba im cc tr ú lp thnh mt tam giỏc cú mt gúc bng 1200 x Ta cú y x 4mx ; y x( x m) x m (m < 0) Khi ú cỏc im cc tr l: A(0; m2 m), B m; m , C m; m uur uuur AB ( m; m2 ) ; AC ( m; m2 ) ABC cõn ti A nờn gúc 120o chớnh l A uur uuur AB.AC m m m A 120o cos A uur uuur 2 m4 m AB AC m (loaùi) 4 2m 2m m m 3m m m m4 m Vy m 3 m m4 Cõu 32 Cho hm s y x 2mx m cú th (Cm) 1) Kho sỏt s bin thiờn v v th hm s m = 2) Vi nhng giỏ tr no ca m thỡ th (Cm) cú ba im cc tr, ng thi ba im cc tr ú lp thnh mt tam giỏc cú bỏn kớnh ng trũn ngoi tip bng x Ta cú y x 4mx x( x m) x m Hm s ó cho cú ba im cc tr PT y cú ba nghim phõn bit v y i du x i qua cỏc nghim ú m Khi ú ba im cc tr ca th (Cm) l: Trang 10 Gia s Thnh c www.daythem.com.vn a b t3 6t 10t (t 4)(t 2t 2) t (a 1)2 a b Vy im tho YCBT l: A(3;1), B(1; 3) y 3x x (C) 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn ng thng (d): y x cỏc im m t ú k c ỳng tip tuyn phõn bit vi th (C) Cỏc im cn tỡm l: A(2; 2) v B(2; 2) Cõu 65 Cho hm s Cõu 66 Cho hm s y x 3x (C) 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn ng thng (d): y = cỏc im m t ú k c tip tuyn phõn bit vi th (C) Gi M(m;2) (d ) PT ng thng i qua im M v cú h s gúc k cú dng : y k( x m) 2 l tip tuyn ca (C) h PT sau cú nghim x 3x k ( x m) (1) (*) (2) 3x x k Thay (2) v (1) ta c: x3 3(m 1) x 6mx ( x 2) x (3m 1) x x2 f ( x ) x (3m 1) x (3) T M k c tip tuyn n th (C) h (*) cú nghim x phõn bit m m (3) cú hai nghim phõn bit khỏc f (2) m m m Vy t cỏc im M(m; 2) (d): y = vi cú th k c tip tuyn m n (C) mx (m 1) x (4 3m) x cú th l (Cm) 1) Kho sỏt s bin thiờn v v th ca hm s m = 2) Tỡm cỏc giỏ tr m cho trờn th (Cm) tn ti mt im nht cú honh õm m tip tuyn ti ú vuụng gúc vi ng thng (d): x 2y Cõu 67 Cho hm s y f ( x ) (d) cú h s gúc tip tuyn cú h s gúc k Gi x l honh tip im thỡ: f '( x) mx 2(m 1)x (4 3m) mx 2(m 1)x 3m YCBT (1) cú ỳng mt nghim õm + Nu m thỡ (1) x x (loi) 3m + Nu m thỡ d thy phng trỡnh (1) cú nghim l x hay x= m WWW.ToanCaBa.Net Trang 23 (1) m 3m Do ú (1) cú mt nghim õm thỡ m m Vy m hay m Cõu 68 Cho hm s 2 y x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Cho im A(a;0) Tỡm a t A k c tip tuyn phõn bit vi th (C) Ta cú y x x Phng trỡnh ng thng d i qua A(a;0) v cú h s gúc k : y k ( x a) x x k ( x a) d l tip tuyn ca (C) h phng trỡnh sau cú nghim: (I ) x3 x k k x( x 1) k ( A) Ta cú: (I ) hoc ( B) f ( x ) x ax (1) x + T h (A), ch cho ta mt tip tuyn nht l d1 : y + Vy t A k c tip tuyn phõn bit vi (C) thỡ iu kin cn v l h (B) phi cú nghim phõn bit ( x; k ) vi x , tc l phng trỡnh (1) phi cú nghim phõn 3 bit khỏc 4a a a 2 f (1) Cõu 69 Cho hm s y f ( x ) x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Trờn (C) ly hai im phõn bit A v B cú honh ln lt l a v b Tỡm iu kin i vi a v b hai tip tuyn ca (C) ti A v B song song vi Ta cú: f '( x) x3 x H s gúc tip tuyn ca (C) ti A v B l k A f '(a) 4a3 4a, kB f '(b) 4b3 4b Tip tuyn ti A, B ln lt cú phng trỡnh l: y f (a)( x a) f (a) y f (a) x f (a) af (a) y f (b)( x b) f (b) y f (b) x f (b) bf (b) Hai tip tuyn ca (C) ti A v B song song hoc trựng v ch khi: k A kB 4a3 4a = 4b3 4b (a b)(a2 ab b2 1) (1) Vỡ A v B phõn bit nờn a b , ú (1) a2 ab b2 (2) Mt khỏc hai tip tuyn ca (C) ti A v B trựng v ch khi: a2 ab b2 a2 ab b2 ( a b) 4 3a 2a 3b 2b f (a) af (a) f (b) bf (b) Gii h ny ta c nghim l (a; b) (1;1) hoc (a; b) (1; 1) , hai nghim ny tng ng vi cựng mt cp im trờn th l (1; 1) v (1; 1) Vy iu kin cn v hai tip tuyn ca (C) ti A v B song song vi l: a2 ab b2 a 1; a b Trang 24 Gia s Thnh c www.daythem.com.vn 2x (C) x2 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Vit phng trỡnh tip tuyn ca th (C), bit rng khong cỏch t tõm i xng ca th (C) n tip tuyn l ln nht Tip tuyn (d) ca th (C) ti im M cú honh a thuc (C) cú phng trỡnh: 2a y ( x a) x (a 2)2 y 2a2 a (a 2) Cõu 70 Cho hm s y Tõm i xng ca (C) l I 2;2 Ta cú: d (I , d ) a2 16 (a 2) a2 2.4.(a 2) a2 2 a2 2 a d (I , d ) ln nht (a 2)2 a T ú suy cú hai tip tuyn y x v y x x2 (1) 2x 1) Kho sỏt s bin thiờn v v th ca hm s (1) 2) Vit phng trỡnh tip tuyn ca th hm s (1), bit tip tuyn ú ct trc honh, trc tung ln lt ti hai im phõn bit A, B v tam giỏc OAB cõn ti gc ta O Gi ( x0 ; y0 ) l to ca tip im y ( x0 ) (2 x0 3)2 Cõu 71 Cho hm s y OAB cõn ti O nờn tip tuyn song song vi ng thng y x (vỡ tip tuyn cú h s x0 y0 1 gúc õm) Ngha l: y ( x0 ) (2 x0 3)2 x0 y0 + Vi x0 1; y0 : y ( x 1) y x (loi) + Vi x0 2; y0 : y ( x 2) y x (nhn) Vy phng trỡnh tip tuyn cn tỡm l: y x 2x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Lp phng trỡnh tip tuyn ca th (C) cho tip tuyn ny ct cỏc trc Ox, Oy ln lt ti cỏc im A v B tho OA = 4OB Gi s tip tuyn d ca (C) ti M ( x0 ; y0 ) (C) ct Ox ti A, Oy ti B cho OA 4OB Cõu 72 Cho hm s y = OB 1 H s gúc ca d bng hoc OA 4 x ( y ) 0 1 H s gúc ca d l y ( x0 ) ( x0 1)2 ( x0 1)2 x (y ) Do OAB vuụng ti O nờn tan A WWW.ToanCaBa.Net Trang 25 y ( x 1) y x Khi ú cú tip tuyn tho l: 13 y ( x 3) y x 4 2x cú th (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn (C) nhng im M cho tip tuyn ti M ca (C) ct hai tim cn ca (C) ti A, B cho AB ngn nht 1 Ly im M m; C Ta cú: y (m) m2 (m 2)2 Cõu 73 Cho hm s y Tip tuyn (d) ti M cú phng trỡnh: y ( x m) (m 2) Giao im ca (d) vi tim cn ng l: A 2;2 m2 Giao im ca (d) vi tim cn ngang l: B(2m 2;2) m2 m Ta cú: AB2 (m 2)2 Du = xy m (m 2) Vy im M cn tỡm cú ta l: M(3;3) hoc M(1;1) 2x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Cho M l im bt kỡ trờn (C) Tip tuyn ca (C) ti M ct cỏc ng tim cn ca (C) ti A v B Gi I l giao im ca cỏc ng tim cn Tỡm to im M cho ng trũn ngoi tip tam giỏc IAB cú din tớch nh nht Cõu 74 Cho hm s y 2x Gi s M x0 ; , x0 , y '( x0 ) x0 x0 Phng trỡnh tip tuyn () vi ( C) ti M: y x0 ( x x0 ) x0 x0 2x To giao im A, B ca () vi hai tim cn l: A 2; x ; B x0 2;2 y y 2x x x B x0 yM suy M l trung im Ta thy A x0 x M , A B x0 2 ca AB Mt khỏc I(2; 2) v IAB vuụng ti I nờn ng trũn ngoi tip tam giỏc IAB cú din tớch x0 2 ( x0 2)2 S = IM ( x0 2) x0 ( x0 2)2 x ( x0 2) x0 Do ú im M cn tỡm l M(1; 1) hoc M(3; 3) Du = xy ( x0 2)2 Trang 26 Gia s Thnh c www.daythem.com.vn 2x cú th (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Gi I l giao im ca hai tim cn Tỡm im M thuc (C) cho tip tuyn ca (C) ti M ct tim cn ti A v B vi chu vi tam giỏc IAB t giỏ tr nh nht Cõu 75 Cho hm s y (C) Giao im ca tim cn l I(1;2) Gi M x0 ;2 x0 3 ( x x0 ) + PTTT ti M cú dng: y x0 ( x 1) + To cỏc giao im ca tip tuyn vi tim cn: A 1;2 + Ta cú: SIAB , B (2 x 1;2) x0 1 IA.IB x0 2.3 (vdt) 2 x0 + IAB vuụng cú din tớch khụng i chu vi IAB t giỏ tr nh nht IA= IB x0 x0 x0 x0 Vy cú hai im M tha iu kin M1 3;2 , M2 3;2 Khi ú chu vi AIB = Chỳ ý: Vi s dng a, b tho ab = S (khụng i) thỡ biu thc P = a b a2 b2 nh nht v ch a = b Tht vy: P = a b a2 b2 ab 2ab (2 2) ab (2 2) S Du "=" xy a = b x2 (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Cho im A(0; a) Tỡm a t A k c tip tuyn ti th (C) cho tip im tng ng nm v phớa ca trc honh Phng trỡnh ng thng d i qua A(0; a) v cú h s gúc k: y kx a Cõu 76 Cho hm s: y x x kx a d l tip tuyn ca (C) H PT cú nghim k ( x 1)2 PT: (1 a) x 2(a 2)x (a 2) (1) cú nghim x qua A cú tip tuyn thỡ (1) phi cú nghim phõn bit x1, x2 a a a 3a Khi ú ta cú: x1 x2 (*) 3 2(a 2) a2 ; x1x2 ; y2 v y1 a a x1 x2 tip im nm v phớa i vi trc honh thỡ y1.y2 WWW.ToanCaBa.Net Trang 27 x1.x2 2( x1 x2 ) 3a a x1.x2 ( x1 x2 ) x1 x2 Kt hp vi iu kin (*) ta c: a a x Cõu 77 Cho hm s y x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Cho im Mo ( xo ; yo ) thuc th (C) Tip tuyn ca (C) ti M0 ct cỏc tim cn ca (C) ti cỏc im A v B Chng minh Mo l trung im ca on thng AB Mo ( xo ; yo ) (C) y0 x0 Phng trỡnh tip tuyn (d) ti M0 : y y0 ( x0 1)2 ( x x0 ) Giao im ca (d) vi cỏc tim cn l: A(2 x0 1;1), B(1;2 y0 1) x A xB y y x0 ; A B y0 M0 l trung im AB 2 x2 (C) x 1) Kho sỏt s bin thiờn 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 giỏc cú din tớch khụng i a2 Gi s M a; (C) a Cõu 78 Cho hm s : y PTTT (d) ca (C) ti M: y y (a).( x a) a2 4a a2 x y a (a 1)2 (a 1) a5 Cỏc giao im ca (d) vi cỏc tim cn l: A 1; , B(2a 1;1) a IA 0; ; IB (2a 2; 0) IB a IA a a Din tớch IAB : S IAB = IA.IB = (vdt) PCM 2x Cõu hi tng t i vi hm s y S: S = 12 x x2 x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Gi I l giao im ca ng tim cn, l mt tip tuyn bt k ca th (C) d l khong cỏch t I n Tỡm giỏ tr ln nht ca d Cõu 79 Cho hm s y = y x Giao im ca hai ng tim cn l I(1; 1) Gi s M x0 ; (C ) x0 ( x 1)2 Trang 28 Gia s Thnh c www.daythem.com.vn Phng trỡnh tip tuyn vi thi hm s ti M l: x 2 y ( x x0 ) x x0 y x0 x0 x0 x0 x0 Khong cỏch t I n l d = Vy GTLN ca d bng x0 1 x0 = x0 x0 2 x0 hoc x0 2x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Vit phng trỡnh tip tuyn ca (C), bit khong cỏch t im I(1; 2) n tip tuyn bng Cõu 80 Cho hm s y Tip tuyn ca (C) ti im M ( x0 ; f ( x0 )) (C) cú phng trỡnh: y f '( x0 )( x x0 ) f ( x0 ) x ( x0 1)2 y x02 x0 (*) Khong cỏch t im I(1; 2) n tip tuyn (*) bng Cỏc tip tuyn cn tỡm : x y v x y x0 x x0 ( x0 1)4 x (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn Oy tt c cỏc im t ú k c nht mt tip tuyn ti (C) Gi M (0; yo ) l im cn tỡm PT ng thng qua M cú dng: y kx yo (d) Cõu 81 Cho hm s y x ( y 1) x 2( y 1) x y (1) kx yo o o x o (d) l tip tuyn ca (C) (*) k k x 1; ( x 1) ( x 1)2 YCBT h (*) cú 1nghim (1) cú nghim khỏc yo yo x ; yo k 2 x ' ( yo 1) ( yo 1)( yo 1) x 0; yo k Vy cú im cn tỡm l: M(0; 1) v M(0; 1) 2x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Vit phng trỡnh tip tuyn ca th (C), bit rng tip tuyn cỏch u hai im A(2; 4), B(4; 2) Gi x0 l honh tip im ( x0 ) Cõu 82 Cho hm s y WWW.ToanCaBa.Net Trang 29 PTTT (d) l y ( x0 1) ( x x0 ) x0 x0 x ( x0 1)2 y x02 x0 Ta cú: d ( A, d ) d (B, d ) 4( x0 1)2 x02 x0 2( x0 1)2 x02 x0 x0 x0 x0 Vy cú ba phng trỡnh tip tuyn: y x ; y x 1; y x 4 2x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Gi I l giao im ca hai ng tim cn, A l im trờn (C) cú 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 tớnh din tớch tam giỏc IPQ 2a 2a I (1; 2), A a; ( x a) PT tip tuyn d ti A: y (1 a) a a 2a Giao im ca tim cn ng v tip tuyn d: P 1; a Giao im ca tim cn ngang v tip tuyn d: Q(2a 1; 2) Ta cú: xP xQ 2a x A Vy A l trung im ca PQ Cõu 83 Cho hm s y IP = SIPQ = 2a 2 ; IQ = 2(a 1) a a IP.IQ = (vdt) 2x (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Vit phng trỡnh tip tuyn ti im M thuc (C) bit tip tuyn ú ct tim cn ng v ã tim cn ngang ln lt ti A, B cho cụsin gúc ABI bng , vi I l giao tim cn 17 Cõu 84 Cho hm s y I(2; 2) Gi M x0 ; x0 (C ) , x0 x0 Phng trỡnh tip tuyn ti M: y ( x0 2) ( x x0 ) x0 x0 2x Giao im ca vi cỏc tim cn: A 2; , B(2 x0 2;2) x0 ã ã x IA Do cos ABI nờn tan ABI IB2 16.IA2 ( x0 2)4 16 IB 17 x0 3 Kt lun: Ti M 0; phng trỡnh tip tuyn: y x Ti M 4; phng trỡnh tip tuyn: y x Trang 30 Gia s Thnh c www.daythem.com.vn KSHS 05: BIN LUN S NGHIM CA PHNG TRèNH Cõu 85 Cho hm s y x 3x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm m phng trỡnh x3 3x m3 3m2 cú ba nghim phõn bit PT x3 3x m3 3m2 x3 3x m3 3m2 t k m3 3m2 S nghim ca PT bng s giao im ca th (C) vi ng thng d: y k Da vo th (C) ta cú PT cú nghim phõn bit k m (1;3) \ {0;2} Cõu 86 Cho hm s y x 5x cú th (C) 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm m phng trỡnh | x4 5x | log m cú nghim Da vo th ta cú PT cú nghim log12 m 9 m 12 144 12 Cõu 87 Cho hm s: y x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Bin lun theo m s nghim ca phng trỡnh: x x log2 m x x log2 m x x log2 m (m > 0) (*) + S nghim ca (*) l s giao im ca th y x x v y log2 m + T th suy ra: 1 m m m m 0m 2 2 nghim nghim nghim nghim vụ nghim Cõu 88 Cho hm s y f ( x ) 8x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Da vo th (C) hóy bin lun theo m s nghim ca phng trỡnh: 8cos4 x 9cos2 x m vi x [0; ] Xột phng trỡnh: 8cos4 x 9cos2 x m vi x [0; ] (1) t t cos x , phng trỡnh (1) tr thnh: 8t 9t m (2) Vỡ x [0; ] nờn t [1;1] , gia x v t cú s tng ng mt i mt, ú s nghim ca phng trỡnh (1) v (2) bng Ta cú: (2) 8t 9t m (3) Gi (C1): y 8t 9t vi t [1;1] v (d): y m Phng trỡnh (3) l phng trỡnh honh giao im ca (C1) v (d) Chỳ ý rng (C1) ging nh th (C) x WWW.ToanCaBa.Net Trang 31 Da vo th ta cú kt lun sau: m0 m0 m vụ nghim nghim nghim 81 32 nghim m 81 32 nghim 81 32 vụ nghim m m 3x (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s Cõu 89 Cho hm s y 2) Tỡm cỏc giỏ tr ca m phng trỡnh sau cú nghim trờn on 0; : sin6 x cos6 x m (sin4 x cos4 x) Xột phng trỡnh: sin6 x cos6 x m (sin4 x cos4 x) (*) sin2 x m sin2 x 3sin2 x 2m(2 sin2 x) t t sin2 x Vi x 0; thỡ t 0;1 Khi ú (1) tr thnh: 3t vi t 0;1 2m t sin x t Nhn xột : vi mi t 0;1 ta cú : sin x t sin x t (*) cú nghim thuc on 0; thỡ t ;1 t ;1 7 Da vo th (C) ta cú: y(1) 2m y 2m m 10 x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s x m 2) Bin lun theo m s nghim ca phng trỡnh x Cõu 90 Cho hm s y S nghim ca x m bng s giao im ca th (C): y x Da vo th ta suy c: m 1; m nghim m 1 nghim x x v y m m vụ nghim KSHS 06: IM C BIT CA TH Cõu 91 Cho hm s y x3 3x (C) 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm im trờn th hm s cho chỳng i xng qua tõm M(1; 3) Trang 32 (1) Gia s Thnh c www.daythem.com.vn Gi A x0 ; y0 , B l im i xng vi A qua im M(1;3) B x0 ;6 y0 y x0 x0 A, B (C ) y0 (2 x0 ) 3(2 x0 ) x03 3x0 x0 x0 x02 12 x0 x0 y0 Vy im cn tỡm l: 1; v 1;6 Cõu 92 Cho hm s y x3 3x (C) 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn (C) hai im i xng qua ng thng d: x y Gi M x1; y1 ; N x2 ; y2 thuc (C) l hai im i xng qua ng thng d x x y y I l trung im ca AB nờn I ; , ta cú I d 3 y1 y2 x1 3x1 x2 3x2 x x Cú: 2 2 2 x1 x2 x1 x2 3x1 x2 x1 x2 x1 x2 x1 x2 2 x1 x1 x2 x2 Li cú: MN d x2 x1 y2 y1 x2 x1 x2 x1 x12 x1 x2 x22 x12 x1 x2 x22 7 ; x2 2 x12 x1 x2 x22 x1 x22 - Xột vụ nghim x1 x1 x2 x2 x x 7 7 Vy im cn tỡm l: ;2 ; ; 2 2 - Xột x1 x2 x1 x3 11 x 3x Cõu 93 Cho hm s y 3 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn th (C) hai im phõn bit M, N i xng qua trc tung x x1 Hai im M ( x1; y1 ), N ( x2 ; y2 ) (C) i xng qua Oy y1 y2 x2 x1 x1 x1 x3 hoc x 11 11 x2 x2 x12 3x1 x23 3x 3 3 16 16 Vy hai im thuc th (C) v i xng qua Oy l: M 3; , N 3; WWW.ToanCaBa.Net Trang 33 2x (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm im M thuc th (C) tip tuyn ca (C) ti M vi ng thng i qua M v giao im hai ng tim cn cú tớch cỏc h s gúc bng Giao im tim cn l I(1;2) Cõu 94 Cho hm s y y M yI Gi M x0 ;2 (C ) kIM x0 x M xI ( x 1)2 + H s gúc ca tip tuyn ti M: kM y ( x0 ) x0 x + YCBT kM kIM Vy cú im M tha món: M(0; 3) v M(2; 5) x0 2x (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn (C) nhng im cú tng khong cỏch n hai tim cn ca (C) nh nht 2x 1 Gi M ( x0 ; y0 ) (C), ( x0 ) thỡ y0 x0 x0 Gi A, B ln lt l hỡnh chiu ca M trờn TC v TCN thỡ: MA x0 , MB y0 x0 Cõu 95 Cho hm s y p dng BT Cụ-si ta cú: MA MB MA.MB x0 x0 x x0 x0 Vy ta cú hai im cn tỡm l (0; 1) v (2; 3) Cõu hi tng t: 2x a) y S: x0 x MA + MB nh nht bng x0 3x (C) x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm cỏc im thuc (C) cỏch u tim cn Gi M ( x; y) (C) v cỏch u tim cn x = v y = Cõu 96 Cho hm s y 3x x x x x ( x 2) x x x x Vy cú im tho bi l : M1( 1; 1) v M2(4; 6) Ta cú: x y x 2x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s Cõu 97 Cho hm s y Trang 34 Gia s Thnh c www.daythem.com.vn 2) Tỡm trờn (C) hai im i xng qua ng thng MN bit M(3; 0) v N(1; 1) uuur MN (2; 1) Phng trỡnh MN: x 2y Phng trỡnh ng thng (d) MN cú dng: y x m Phng trỡnh honh giao im ca (C) v (d): 2x (1) x m x mx m ( x 1) x (d) ct (C) ti hai im phõn bit A, B m2 8m 32 (2) Khi ú A( x1;2 x1 m), B( x2 ;2 x2 m) vi x1, x2 l cỏc nghim ca (1) x x m m Trung im ca AB l I ; x1 x2 m I ; (theo nh lý Vi-et) A, B i xng qua MN I MN m x Suy (1) x x A(0; 4), B(2; 0) x Cõu 98 Cho hm s y 2x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn th (C) hai im B, C thuc hai nhỏnh cho tam giỏc ABC vuụng cõn ti nh A vi A(2; 0) Ta cú (C ) : y Gi B b; , C c; vi b c b c x Gi H, K ln lt l hỡnh chiu ca B, C lờn trc Ox ã ã ã ã ã ã ã Ta cú: AB AC; BAC 900 CAK BAH 900 CAK ACK BAH ACK ã ã AH CK v: BHA CKA 900 ABH CAK C HB AK b c b Hay: c c2 b Vy B(1;1), C(3;3) B H A K 2x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm ta im M (C) cho khong cỏch t im I (1; 2) ti tip tuyn ca (C) ti M l ln nht Cõu 99 Cho hm s y (C ) PTTT ca (C) ti M l: Gi s M x0 ; x 3 ( x x0 ) 3( x x0 ) ( x0 1)2 ( y 2) 3( x0 1) x0 ( x0 1) Khong cỏch t I (1;2) ti tip tuyn l: y2 WWW.ToanCaBa.Net Trang 35 d 3(1 x0 ) 3( x0 1) x0 x0 ( x0 1) ( x0 1) 2 ( x0 1) ( x0 1) d ( x0 1) Khong cỏch d ln nht bng ( x0 1)2 x0 x0 ( x0 1) Theo BT Cụsi: Vy cú hai im cn tỡm l: M ;2 hoc M ;2 x2 2x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm nhng im trờn th (C) cỏch u hai im A(2; 0) v B(0; 2) PT ng trung trc an AB: y x Nhng im thuc th cỏch u A v B cú honh l nghim ca PT: x x2 x x2 x 2x 1 x Cõu 100 Cho hm s y 5 5 Hai im cn tỡm l: , , ; 2 x x 1) Kho sỏt s bin thiờn v v th (C) ca hm s 2) Tỡm trờn hai nhỏnh ca th (C) hai im A v B cho AB ngn nht Tp xỏc nh D = R \ { 1} Tim cn ng x Cõu 101 Cho hm s y Gi s A a;1 , B b;1 (vi a 0, b ) l im thuc nhỏnh ca (C) a b 1 16 16 64 AB (a b) 16 (a b)2 4ab 4ab 32 ab a b a2 b2 a2 b2 2 a b a b AB nh nht AB ab44 16 4ab a ab Khi ú: A 4;1 64 , B 4;1 64 Trang 36 Gia s Thnh c WWW.ToanCaBa.Net www.daythem.com.vn Trang 37 [...]... thi n v v th ca hm s khi m = 3 2) Tỡm m th (Cm) ct trc honh ti mt im duy nht Phng trỡnh honh giao im ca (Cm) vi trc honh: 2 x3 mx 2 0 m x 2 ( x 0) x Xột hm s: f ( x ) x 2 Ta cú bng bin thi n: 2 2 2 x 3 2 f '( x ) 2 x x x2 x2 x 0 1 + f (x) + 0 3 f (x) th (Cm) ct trc honh ti mt im duy nht m 3 Cõu 47 Cho hm s y 2 x 3 3(m 1) x 2 6mx 2 cú th (Cm) 1) Kho sỏt s bin thi n... (1) cú ỳng mt nghim õm + Nu m 0 thỡ (1) 2 x 2 x 1 (loi) 2 3m + Nu m 0 thỡ d thy phng trỡnh (1) cú 2 nghim l x 1 hay x= m WWW.ToanCaBa.Net Trang 23 (1) m 0 2 3m Do ú (1) cú mt nghim õm thỡ 0 m 2 m 3 2 Vy m 0 hay m 3 Cõu 68 Cho hm s 2 2 y x 1 x 1 1) Kho sỏt s bin thi n v v th (C) ca hm s 2) Cho im A(a;0) Tỡm a t A k c 3 tip tuyn phõn bit vi th (C) Ta cú y x 4 2 x 2 1 Phng... x2 x2 x3 x1 x3 m 1 x x x 2 1 2 3 Vỡ x1 x3 x22 x23 2 x2 3 2 nờn ta cú: m 1 4 3 2.3m m k : Vi m Vy m 5 3 2 1 3 5 , thay vo tớnh nghim thy tha món 33 2 1 5 3 2 1 3 Cõu 43 Cho hm s y x 3 2mx 2 (m 3) x 4 cú th l (Cm) (m l tham s) 1) Kho sỏt s bin thi n v v th (C1) ca hm s trờn khi m = 1 2) Cho ng thng (d): y x 4 v im K(1; 3) Tỡm cỏc giỏ tr ca m (d) ct (Cm) ti ba im phõn bit... 2 m 5 1 4SV ABC 4m m 2 Cõu hi tng t: SV ABC a) y x 4 2mx 2 1 S: m 1, m 1 5 2 Cõu 33 Cho hm s y x 4 2mx 2 2m m4 cú th (Cm) 1) Kho sỏt s bin thi n v v th hm s khi m = 1 2) Vi nhng giỏ tr no ca m thỡ th (Cm) cú ba im cc tr, ng thi ba im cc tr ú lp thnh mt tam giỏc cú din tớch bng 4 x 0 Ta cú y ' 4 x3 4mx 0 2 g ( x) x m 0 Hm s cú 3 cc tr y ' 0 cú 3 nghim phõn bit g m... cú th l (C) 1) Kho sỏt s bin thi n v v th (C) ca hm s 2) nh m ng thng (d ) : y mx 2m 4 ct th (C) ti ba im phõn bit PT honh giao im ca (C) v (d): x3 6 x 2 9x 6 mx 2m 4 x 2 ( x 2)( x 2 4 x 1 m) 0 2 g( x ) x 4 x 1 m 0 (d) ct (C) ti ba im phõn bit PT g( x ) 0 cú 2 nghim phõn bit khỏc 2 m 3 Cõu 49 Cho hm s y x 3 3x 2 1 1) Kho sỏt s bin thi n v v th (C) ca hm s 2) Tỡm... (loi) + y(m) 0 2m3 2m 0 m 0 m 1 Vy: m 1 Cõu 51 Cho hm s y x 4 mx 2 m 1 cú th l Cm 1) Kho sỏt s bin thi n v v th (C) ca hm s khi m 8 2) nh m th Cm ct trc trc honh ti bn im phõn bit m 1 m 2 Cõu 52 Cho hm s y x 4 2 m 1 x 2 2m 1 cú th l Cm 1) Kho sỏt s bin thi n v v th ca hm s ó cho khi m 0 2) nh m th Cm ct trc honh ti 4 im phõn bit cú honh lp thnh cp s cng Xột... giao im ca d v th hm s (1) l A( x1; x1 2), B( x2 ; x2 2) Suy ra AB2 2( x1 x2 )2 2 ( x1 x2 )2 4 x1x2 2(m2 6m 3) m 1 Theo gi thit ta c 2(m2 6m 3) 8 m2 6m 7 0 m 7 Kt hp vi iu kin (**) ta c m 7 l giỏ tr cn tỡm 2x 1 (C) x 1 1) Kho sỏt s bin thi n v v th (C) ca hm s 2) Tỡm m ng thng d: y x m ct (C) ti hai im phõn bit A, B sao cho OAB vuụng ti O Cõu 61 Cho hm s y Phng trỡnh... 1)2 4 a 1 b 3 Vy 2 im tho món YCBT l: A(3;1), B(1; 3) y 3x x 3 (C) 1) Kho sỏt s bin thi n v v th (C) ca hm s 2) Tỡm trờn ng thng (d): y x cỏc im m t ú k c ỳng 2 tip tuyn phõn bit vi th (C) Cỏc im cn tỡm l: A(2; 2) v B(2; 2) Cõu 65 Cho hm s Cõu 66 Cho hm s y x 3 3x 2 2 (C) 1) Kho sỏt s bin thi n v v th (C) ca hm s 2) Tỡm trờn ng thng (d): y = 2 cỏc im m t ú k c 3 tip tuyn phõn bit... tip tuyn ti M v N vuụng gúc vi nhau y ( xM ).y ( xN ) 1 (3xM2 6 xM )(3xN2 6 xN ) 1 9k 2 18k 1 0 k 3 2 2 3 (tho (*)) Cõu 37 Cho hm s y x 3 3x (C) 1) Kho sỏt s bin thi n v v th (C) ca hm s 2) Chng minh rng khi m thay i, ng thng (d): y m( x 1) 2 luụn ct th (C) ti mt im M c nh v xỏc nh cỏc giỏ tr ca m (d) ct (C) ti 3 im phõn bit M, N, P sao cho tip tuyn ca (C) ti N v P vuụng gúc vi nhau... bin thi n v v th (C) ca hm s 2) Cho im Mo ( xo ; yo ) thuc th (C) Tip tuyn ca (C) ti M0 ct cỏc tim cn ca (C) ti cỏc im A v B Chng minh Mo l trung im ca on thng AB 4 Mo ( xo ; yo ) (C) y0 1 x0 1 Phng trỡnh tip tuyn (d) ti M0 : y y0 4 ( x0 1)2 ( x x0 ) Giao im ca (d) vi cỏc tim cn l: A(2 x0 1;1), B(1;2 y0 1) x A xB y y x0 ; A B y0 M0 l trung im AB 2 2 x2 (C) x 1 1) Kho sỏt s bin thi n