PHẦN 1 đồ THỊ hàm số

www.TOANTUYENSINH.com PHN TH HM S 1.1 S ng bin nghch bin ca hm s Cõu Cho hm s y = (m m) x + 2mx + x Tỡm m hm s luụn ng bin trờn Ă Tp xỏc nh: D = Ă o hm: y ' = (m m) x + 4mx + Hm s luụn ng bin trờn Ă y ' x Ă m = m = Trng hp 1: Xột m m = + Vi m = , ta cú y ' = > 0, x Ă , suy m = tha + Vi m = , ta cú y ' = x + > x > , suy m = khụng tha m , ú: m Trng hp 2: Xột m m ' = m + 3m m y ' x Ă m < m m > m < m > T hai trng hp trờn, ta cú giỏ tr m cn tỡm l m Cõu Cho hm s y = x 3mx + 3(m 1)x 2m + Tỡm m hm s nghch bin trờn khong ( 1;2 ) Tp xỏc nh: D = Ă o hm: y ' = 3x 6mx + 3(m 1) Hm s nghch bin trờn khong ( 1;2 ) y ' x ( 1; ) Ta cú ' = 9m 9(m 1) = > 0, m Suy y ' luụn cú hai nghim phõn bit x1 = m 1; x2 = m + ( x1 < x2 ) x1 m m m + x2 Do ú: y ' x ( 1; ) x1 < x2 Vy giỏ tr m cn tỡm l m Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Cõu Xỏc nh m hm s sau ng bin khong (0; +): x +m y= x +1 + TX: D = R + y = mx + ( x + 1) x + Hm s B (0; +) y mi x (0; +) -mx + mi x (0; +) (1) m = (1) ỳng m > : -mx + x 1/m Vy (1) khụng tha m < 0: -mx + x 1/m Khi ú (1) 1/m t/m Giỏ tr cn tỡm l: m Cõu Cho hm s y = x 3x mx + Tỡm m hm s ng bin trờn khong ( 0; + ) Tp xỏc nh: D = Ă o hm: y ' = 3x x m Hm s ng bin trờn khong ( 0; + ) y ' , x ( 0; + ) (cú du bng) x x m , x ( 0; + ) x x m , x ( 0; + ) (*) Xột hm s f ( x) = x - x , x ( 0; + ) , ta cú: f '( x ) = x - ; f '( x) = x = Bng bin thiờn: x +Ơ f '( x) f ( x) - + +Ơ - T BBT ta suy ra: (*) m Ê - Vy giỏ tr m cn tỡm l m Ê - Cõu Tỡm m hm sụluụn nghch biờn: y = x3 + (3 m) x 2mx + 12 + Tp xỏc nh: D = Ă Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com + o hm: y ' = 3x + 2(3 m) x 2m + hm sụluụn nghch biờn thỡ y ' x < a < ' + m 6m (3)(2m) m 12m + 3 m + 3 Cõu Cho hm s y = nh ca nú Tp xỏc nh: D = Ă \ { m} mx + 7m Tỡm m hm s ng bin trờn tng khong xỏc xm m2 7m + o hm: y ' = ( x m) Du ca y ' l du ca biu thc m 7m + Hm s ng bin trờn tng khong xỏc nh y ' > , x D (khụng cú du bng) m m + > - < m * nờn hm s t cc i ti im x = -1 v giỏ tr cc i nờn hm s t cc tiu ti x = v giỏ tr cc tiu Cõu Tỡm cỏc im cc tr ca th hm s y = x3 x + Tỡm cỏc im cc tr ca th hm s y = x x + * Tp xỏc nh: Ă x = y ' = x x, y ' = x = Bng xột du o hm T bng x y Nguyn Vn Lc + + - Ninh Kiu Cn Th + xột u 0933.168.309 www.TOANTUYENSINH.com o hm ta cú Hm sụat cc tai x = v giỏ tri cc y = ; at cc tiờu tai x = v giỏ tri cc tiờu y = Vy im cc i ca th hm s l M ( 0; ) , im cc tiu ca th hm s l N ( 2; ) Cõu Tỡm cỏc im cc tr ca hm s y = x x TX: D = Ă y ' = x -8x = x ( x -1) x D x = y' = x = Bng xột du ca y: x y - - + - -1 0 + + Kt lun: Hm s t cc i ti x = v ycd = y (0) = Hm s t cc tiu ti x = v yct = y ( 1) = Cõu Cho hm s y = x 3mx + ( m 1) x + 2, m l tham s Tỡm tt c cỏc giỏ tr ca m hm s ó cho t cc tiu ti x = Ta cú: y ' = x 6mx + m 1; y '' = x 6m 2 y '(2) = y ''(2) > Hm s ó cho t cc tiu ti x = m 12m + 11 = m =1 12 6m > Vy vi m = thỡ tha yờu cu bi toỏn Cõu 23 Cho hm s y = x3 3mx + 3(m 1) x m + m (1) 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 Ta cú y = x 6mx + 3(m 1) Hm s (1) cú cc tr thỡ PT y = cú nghim phõn bit x 2mx + m = cú nhim phõn bit = > 0, m Khi ú, im cc i A(m 1;2 2m) v im cc tiu B(m + 1; 2m) Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com m = + 2 OA = OB m + m + = Ta cú m = 2 y= Cõu Tỡm m hm s y= Tỡm m hm s m ( x 1) + ( m ) x t cc tiu ti im x = m ( x 1) + ( m ) x t cc tiu ti im x = y ' = m ( x 1) + m iu kin cn y ' ( 1) = m = Th li m = : y ' = ( x 1) i du t õm sang dng i qua x = Vy nhn m = 2 Cõu Tỡm m hm s: y = 13 x + ( m m + ) x + ( 3m + 1) x + m t cc tiu ti x = y ( x ) = x + ( m m + ) x + 3m + y ( x ) = x + ( m m + ) hm s t cc tiu ti x = thỡ y ( ) = ( m 1) ( m 3) = m + m = m=3 ( ) m m > m m > y ( ) > Cõu Cho hm s y = x 3(m + 1) x + x m , vi m l tham s thc Xỏc nh m hm s ó cho t cc tr ti x1 , x cho x1 x2 = x x =2 Xỏc nh m hm s ó cho t cc tr ti x1 , x cho Ta cú: y ' = 3x 6(m + 1) x + Hm s t cc i, cc tiu ti x1 , x Phng trỡnh y ' = cú hai nghim pb l x1 , x Pt x 2(m + 1) x + = cú hai nghim phõn bit l x1 , x ' = (m + 1) > m > + (1) m < Vi K (1), theo nh lý Viet ta cú: x1 + x = 2(m + 1); x1 x = x1 x2 = ( x1 + x2 ) x1 x2 = ( m + 1) 12 = Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com (m + 1) = m = m = (2) m = T (1) v (2) ta c: m = TMYCBT Cõu Cho hm s: y = x 3( m + 1) x + x m , vi m l tham s thc.Xỏc nh m hm s ó cho t cc tr ti x1 , x2 cho x1 x2 Ta cú y ' = x 6(m + 1) x + Hm s cú cc i, cc tiu x1, x2 PT y = cú hai nghim phõn bit l x1, x2 x 2(m + 1) x + = cú hai nghim phõn bit l x1 , x2 ' = (m + 1) > m > + m < (1) Theo ta cú: x1 x2 ( x1 + x2 ) x1 x2 (*) Theo nh lý Viet ta cú: x1 + x2 = 2(m + 1); x1 x2 = (*) ( m + 1) 12 (m + 1) m (2) T (1) v (2) suy giỏ tr m cn tỡm l: m < hoc + < m Cõu 10 m hm s f ( x ) = 13 mx ( m 1) x + ( m ) x + 13 t cc tr ti x1, x2 tha x1 + x2 = Hm s cú C, CT f ( x ) = mx ( m 1) x + ( m ) = cú nghim phõn bit { m = 0( m 1) 3m ( m 2) > < m < + (*) 2 Vi iu kin (*) thỡ f ( x ) = cú nghim phõn bit x1, x2 v hm s f (x) t cc tr ( m 1) ; x x = ( m ) ti x1, x2 Theo nh lý Viet ta cú: x1 + x2 = m m ( ) ( ) Ta cú: x1 + x2 = x2 = m = m ; x1 = m m = 3m m m m m m m = ( ) m ì3m = m ( m ) ( 3m ) = 3m ( m ) m = m m m C giỏ tr ny u tha iu kin (*) Vy x1 + x2 = m = m = 2 Cõu 11 Cho hm s: y = x 2( m + 1) x + (1) Tỡm cỏc giỏ tr ca tham s m hm s (1) cú im cc tr tha giỏ tr cc tiu t giỏ tr ln nht y = 4x3 4(m2+1)x Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com x = y = x = m + hm s (1) luụn cú im cc tr vi mi m 2 xCT = m2 + giỏ tr cc tiu yCT = (m + 1) + Vỡ (m + 1) yCT max( yCT ) = m + = m = Cõu 12 Cho hm s y = x3 + 3mx + (1) Tỡm m th ca hm s (1) cú im cc tr A, B cho tam giỏc OAB vuụng ti O ( vi O l gc ta ) y ' = x + 3m = ( x m ) y ' = x m = ( *) th hm s (1) cú im cc tr PT (*) cú nghim phõn bit m > ( **) ( ) Khi ú im cc tr A m ;1 2m m , B uuu r uuu r ( m ;1 + 2m m ) Tam giỏc OAB vuụng ti O OA.OB = 4m + m = m = Vy m = ( TM (**) ) 2 Cõu 13 Cho hm s f ( x) = x + 2(m 2) x + m 5m + (Cm) Tỡm m (Cm) cú cỏc im cc i, cc tiu to thnh tam giỏc vuụng cõn Hm s cú C, CT m < To cỏc im cc tr l: A(0; m 5m + 5), B ( m ;1 m), C ( m ;1 m) Tam giỏc ABC luụn cõn ti A ABC vuụng ti A m = Cõu 14 Cho hm s y = 2x3 - 3x2 + ( 1) Tỡm ta giao im ca ng thng d : y = 2x + vi th (C) Tỡm ta im M thuc d v cựng vi hai im cc tr ca th (C) to thnh mt tam giỏc vuụng ti M Xột phng trỡnh honh giao im ca d : y = x + v th (C) l: x 3x + = x + x3 x x = (*) Gii phng trỡnh (*) ta c ba nghim phõn bit x1 = 0, x2 = 2, x3 = Vy d ct (C) ti ba im phõn bit A(0;1), B(2;5),C ;0ữ M d : y = 2x + M (t;2t + 1) , ta cỏc im cc tr ca (C) l D(0;1),T (1;0) M u cựng vi hai im cc tr ca th (C) to thnh tam giỏc vuụng ti M uuur uuuu r uuuur uuuu r DM T M = 0(**) , mt khỏc ta cú DM = (t;2t),T M = (t 1;2t + 1) (**) 5t2 + t = t = hoc t = Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com t = M (0;1) D (loi); t = M ; ữ 5 2 Cõu 15 Cho hm s y = x 2m x + ( Cm ) (1) Tỡm m d hm s (1) cú ba im cc tr l ba nh ca mt tam giỏc vuụng cõn x = 2 m (*) Ta cú: y ' = x 4m x = x ( x m ) = 2 x = m Vi iu kin (*) thỡ hm s (1) cú ba im cc tr Gi ba im cc tr l: A ( 0;1) ; B ( m;1 m ) ; C ( m;1 m ) Do ú nu ba im cc tr to thnh mt tam giỏc vuụng cõn, thỡ nh s l A Do tớnh cht ca hm s trựng phng, tam giỏc ABC ó l tam giỏc cõn ri, cho nờn tha iu kin tam giỏc l vuụng, thỡ AB vuụng gúc vi AC uuu r uuur uuur AB = ( m; m ) ; AC = ( m; m ) ; BC = ( 2m;0 ) 2 2 8 Tam giỏc ABC vuụng khi: BC = AB + AC 4m = m + m + ( m + m ) 2m ( m 1) = 0; m = m = Vy vi m = -1 v m = thỡ tha yờu cu bi toỏn Cõu 16 Cho hm s y = x 2m x + (1).Tỡm tt c cỏc giỏ tr m th hm s (1) cú ba im cc tr A, B, C v din tớch tam giỏc ABC bng 32 (n v din tớch) x = +) Ta cú y = 4x3 4m2x ; y = ; K cú im cc tr: m x = m +) Ta ba im cc tr: A(0 ; 1), B(- m ; m4), C(m ; m4) ; +) CM tam giỏc ABC cõn nh A Ta trung im I ca BC l I(0 ; m4) +) SVABC = AI BC = m m = m = 32 m = (tm) Cõu 17 Cho hm s y = x 2mx + m (1), vi m l tham s thc Xỏc nh m hm s (1) cú ba im cc tr, ng thi cỏc im cc tr ca th to thnh mt tam giỏc cú bỏn kớnh ng trũn ngoi tip bng x = y ' = x3 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 hm s l: Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com ( ) ( A ( 0; m 1) , B m ; m + m , C ) m ; m + m 1 yB y A xC xB = m m ; AB = AC = m + m , BC = m m = m4 + m m AB AC.BC R= =1 = m 2m + = m = SVABC 4m m SVABC = ( ) Cõu 18 Cho hm s y = x3 3x2+2 (1) Gi d l ng thng i qua im A(1;1) v cú h s gúc bng Tỡm im M thuc ng thng d tng khong cỏch t M ti hai im cc tr nh nht + d: y=3x-2 + Xột biu thc P=3x-y-2 Thay ta im (0;2)=>P=-4P=6>0 Vy im cc i v cc tiu nm v hai phớa ca ng thng d T õy, MA+MB nh nht => im A, M, B thng hng + Phng trỡnh ng thng AB: y=-2x+2 x= y = 3x + Ta im M l nghim ca h: y = x + y = Cõu 19 Cho hm s y = x x + x (1) Vit phng trỡnh ng thng i qua im A( 1;1 ) v vuụng gúc vi ng thng i qua hai im cc tr ca (C) Viờt phng trỡnh ng thng i qua im A( 1;1 ) v vuụng gúc vi ng thng i qua hai im cc tr ca (C) ung thng i qua c c tr A(1;2) v B(3;-2) l y=-2x+4 Ta cú pt t vuụng gúc vi (AB) nờn cú h s gúc k= ẵ Vy PT ng thng cn tỡm l y = x+ 2 Cõu 20 Cho hm s y = x 3mx + 4m (1), m l tham s Tỡm m th hm s (1) cú hai im cc tr A v B cho im I (1; 0) l trung im ca on AB Ta cú y ' = 3x 6mx x = y ' = 3x 6mx = x = 2m th hm s (1) cú hai cc tr v ch y ' = cú hai nghim phõn bit m Ta cỏc im cc tr l A(0; 4m2 2), B(2m; 4m3 + 4m 2) Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Vy : y d ( I;) = 2a = ( x a ) x (a + 1) y + 2a 4a = (*) a + (a + 1) 4(1) ( a + 1) 2 + 2a 4a + (a + 1) = a +1 + (a + 1) Ta cú: + (a + 1) = 2 + ( a + 1) 2.2( a + 1) + ( a + 1) 2.2( a + 1) = a + d ( I;) a +1 = Vy d ( I ; ) ln nht d ( I ; ) = a +1 a + = a = 22 = (a + 1) C hai giỏ tr u tha a a + = a = + Vi a = thay vo (*) ta c phng trỡnh tip tuyn 4x y = x y = + Vi a = -3 thay vo (*) ta c phng trỡnh tip tuyn x y + 28 = x y + = Túm li: Cú hai tip tuyn cn tỡm cú phng trỡnh l: x y = ; x y = l: l: 2x Tỡm ta im M cho khong cỏch t im x +1 I ( 1; 2) ti tip tuyn ca (C) ti M l ln nht Nu M x0 ; ữ (C ) thỡ tip tuyn ti M cú phng trỡnh x0 + Cõu 12 Cho hm s y = y2+ 3 = ( x x0 ) hay x0 + ( x0 + 1) 3( x x0 ) ( x0 + 1) ( y 2) 3( x0 + 1) = Khong cỏch t I ( 1;2) ti tip tuyn l d= 3(1 x0 ) 3( x0 + 1) + ( x0 + 1) = Theo bt ng thc Cụsi x0 + + ( x0 + 1) = + ( x0 + 1) 2 ( x0 + 1) + ( x0 + 1) = , võy d ( x0 + 1) Khong cỏch d ln nht bng = ( x0 + 1) ( x0 + 1) = x0 = ( x0 + 1) ( Vy cú hai im M: M + 3;2 Nguyn Vn Lc ) ( hoc M 3;2 + Ninh Kiu Cn Th ) 0933.168.309 www.TOANTUYENSINH.com Cõu 13 Cho hm s y = x3 3mx + 3(m 1) x m3 + m (1) 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 Ta cú y = 3x 6mx + 3(m 1) Hm s (1) cú cc tr thỡ PT y = cú nghim phõn bit x 2mx + m = 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 m + 6m + = m = 2 Cõu 14 Cho hm s y = x 3x + m (1) Tỡm m tip tuyn ca th (1) ti im cú honh bng ct cỏc trc Ox, Oy ln lt ti cỏc im A v B cho din tớch tam giỏc OAB bng Vi x0 = y0 = m M(1 ; m 2) - Tip tuyn ti M l d: y = (3x02 x0 )( x x0 ) + m d: y = -3x + m + m+2 - d ct trc Oy ti B: yB = m + B(0 ; m + 2) - d ct trc Ox ti A: = x A + m + x A = - SOAB = m+2 A ; 0ữ 3 m+2 | OA || OB |= | OA || OB |= m + = (m + 2) = 2 m + = m = m + = m = Vy m = v m = - x+2 (C) x a) Kho sỏt s bin thiờn v v th (C) ca hm s b) Chng minh rng mi tip tuyn ca th (C) u lp vi hai ng tim cn mt tam giỏc cú din tớch khụng i a) T lm a+2 b) Gi s M a; ữ (C) a Cõu 15 Cho hm s: y = a + 4a a+2 y = x + PTTT (d) ca (C) ti M: y = y (a ).( x a ) + (a 1) (a 1) a Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com a+5 Cỏc giao im ca (d) vi cỏc tim cn l: A 1; ữ, B(2a 1;1) a IA = 0; ; IB = (2a 2;0) IB = a ữ IA = a a Din tớch IAB : S IAB = IA.IB = (vdt) PCM 2x Cõu 16 Cho hm s y = x2 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 ta im M cho ng trũn ngoi tip tam giỏc IAB cú din tớch nh nht 2x Gi s M x0 ; ữ, x0 , y '( x0 ) = x0 ( x0 ) Phng trỡnh tip tuyn () vi (C) ti M: y = - (x - 2) (x - x0) + 2x0 - x0 - 2x Ta giao im A, B ca () vi hai tim cn l: A 2; ữ; B ( x0 2;2 ) x y + yB x0 x + xB + x0 = = yM suy M l trung im = = x0 = xM , A Ta thy A 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 ữ = ( x0 2) + S = IM = ( x0 2) + ( x0 2) x0 x0 = 1 Du = xy ( x0 2) = ( x0 2) x0 = Do ú im M cn tỡm l M(1; 1) hoc M(3; 3) 2x (C ) tỡm im M (C ) cho tip tuyn ca th x +1 hm s ti M ct hai trc ta ti A, B cho tam giỏc OAB cú din tớch bng Cõu 17 Cho hm s y = Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Gi M ( x0 , y0 ) (C ) y0 = x0 , x0 + y' = ( x + 1) Tip tuyn ti M cú dng: y = y '( x0 )( x x0 ) + y0 y = x0 x02 2 ( x x ) + y = x + (d ) ( x0 + 1) x0 + ( x0 + 1) ( x0 + 1) Gi A = (d ) ox ta im A l nghim ca h: x02 x+ y = ( x0 + 1) ( x0 + 1) y = x = x02 A( x02 ,0) y = Gi B = (d ) oy ta im B l nghim ca h: x02 y = x + ( x0 + 1) ( x0 + 1) x = x = x02 x02 B (0, ) ( x0 + 1) y = ( x0 + 1) Tam giỏc OAB vuụng ti O ; OA = x = x 2 ; OB = x02 x02 = ( x0 + 1) ( x0 + 1) Din tớch tam giỏc OAB: x04 1 = S = OA.OB = ( x0 + 1) x02 = x0 + x02 x0 = x0 = y0 = x = ( x0 + 1) x = x x + x + ( ) 0 x0 = y0 = 1 Vy tỡm c hai im M tha yờu cu bi toỏn: M ( ; 2) ; M (1,1) 2 Cõu 18 Cho hm s y = x 2m x + (1).Tỡm tt c cỏc giỏ tr m th hm s (1) cú ba im cc tr A, B, C v din tớch tam giỏc ABC bng 32 (n v din tớch) x = +) Ta cú y = 4x3 4m2x ; y = ; K cú im cc tr: m x = m +) Ta ba im cc tr: A(0 ; 1), B(- m ; m4), C(m ; m4) ; +) CM tam giỏc ABC cõn nh A Ta trung im I ca BC l I(0 ; m4) +) SVABC = AI BC = m m = m = 32 m = (tm) Cõu 19 Cho hm s y = x 3x + ( C ) Gi d l ng thng i qua im A(- 1; 0) vi h s gúc l k ( k thuc R) Tỡm k ng thng d ct (C) ti ba im phõn bit v hai giao im B, C (B, C khỏc A ) cựng vi gc ta O to thnh mt tam giỏc cú din tớch bng Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com ng thng d i qua A(-1; 0) vi h s gúc l k, cú phng trỡnh l: y = k(x+1) = kx+ k Nu d ct (C) ti ba im phõn bit thỡ phng trỡnh: x3 3x2 + = kx + k x3 3x2 kx + k = (x + 1)( x2 4x + k ) = x = cú ba nghim phõn bit g(x) = x2 4x + k = cú g ( x) = x x + k = ' > k > < k (*) hai nghim phõn bit khỏc - g ( 1) k Vi iu kin: (*) thỡ d ct (C) ti ba im phõn bit A, B, C.Vi A(-1;0), ú B,C cú honh l hai nghim ca phng trỡnh g(x) = Gi B ( x1; y1 ) ; C ( x2 ; y2 ) vi x1; x2 l hai nghim ca phng trỡnh: x x + k = Cũn y1 = kx1 + k ; y2 = kx2 + k uuur Ta cú: BC = ( x2 x1 ; k ( x2 x1 ) ) BC = ( x2 x1 ) Khong cỏch t O n ng thng d: h = (1+ k ) = x 2 x1 ( 1+ k ) k 1+ k Vy theo gi thit: 1 k S = h.BC = k 2 1+ k2 1+ k = k3 = k3 = 1 k3 = k = 4 2x + ( C ) Tỡm tham s m ng thng d: y = - 2x + m ct x +1 th ti hai im phõn bit A, B cho din tớch tam giỏc OAB bng Xột phng trỡnh honh giao iờm ca d v (C): 2x +1 = x + m ( x 1) g ( x) = x (m 4) x + m = (1) x +1 D ct (C) ti iờm phõn bit (1) cú hai nghim phõn bit khỏc -1 Cõu 20 Cho hm s y = = ( m 4) 8(1 m) > m + > m2 + > m R g ( 1) g ( 1) = Chng t vi mi m d luụn ct (C) ti hai im phõn bit A, B Gi A ( x1 ; x1 + m ) ; B ( x2 ; x2 + m ) Vi: x1 , x2 l hai nghim ca phng trỡnh (1) uuu r 2 Ta cú AB = x2 x1 ;2 x1 x2 AB = ( x2 x1 ) + ( x2 x1 ) = x2 x1 ( ( )) Gi H l hỡnh chiu vuụng gúc ca O trờn d, thỡ khong cỏch t O n d l h: m m h= = 22 + Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Theo gi thit: S = Vy: 1 x2 x1 AB.h = 2 5= = m2 + = 2 m + = 42.3 m + = 42.3 m = 40 m = 10 (*) Vi m tha iu kin (*) thỡ d ct (C) ti A, B tha yờu cu bi toỏn Cõu 21 Cho hm s y = x + 2mx + ( m + 3) x + (1) Tỡm m ng thng d: y = x + ct th hm s (1) ti ba im phõn bit A, B, C cho tam giỏc MBC cú din tớch bng (im B, C cú honh khỏc khụng ; M(1;3) ) th (1) ct d ti ba im A, B, C cú honh l nghim ca phng trỡnh: x = x + 2mx + ( m + 3) x + = x + 4; x x + 2mx + m + = x + 2mx + m + = ' = m m > m < m > (*) Vi m tha (*) thỡ d ct (1) ti ba im A(0; 4), cũn hai im B,C cú honh l hai nghim ca phng trỡnh: ' = m m > x + 2mx + m + = m < m > 2; m m + uuur - Ta cú B ( x1 ; x1 + ) ; C ( x2 ; x2 + ) BC = ( x2 x1; x2 x1 ) BC = ( x2 x1 ) + ( x2 x1 ) = x2 x1 2 -Gi H l hỡnh chiu vuụng gúc ca M trờn d h l khong cỏch t M n d thỡ: + 1 h= = S = BC.h = x2 x1 2 = x2 x1 2 - Theo gi thit: S = x2 x1 = 4; ' = 4; m m = m m = Kt lun: vi m tha món: m = m = m = (chn) Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com 1.6 Tng giao th Cõu Cho hm s y = x x + x a) Kho sỏt s bin thiờn v v th (C) ca hm s ó cho b) Tỡm cỏc giỏ tr thc ca tham s m phng trỡnh x 3x + x m = cú mt 2 nghim nht: x = y'= x = Hm s nghch bin trờn cỏc khong(- ;1) v (3;+ ), ng bin trờn khong (1;3) lim y = , lim y = + TX: D = Ă , y / = 3x 12 x + x BBT x + x y' y + + + + -1 th : i qua cỏc im (3;-1), (1;3), (2;1), (0;-1) Pt : x 3x + x m = x3 x + x = 2m (*) 2 Pt (*) l pt honh giao im ca (C) v ng thng d y = 2m (d cựng phng trc Ox) S nghim ca phng trỡnh l s giao im ca (C) v d Da vo th (C), m < pt cú mt nghim nht thỡ : 2m > m < m > Cõu Cho hm s y = x3 6x2 + 9x (1) a) Kho sỏt s bin thiờn v v th (C) ca hm s (1) b) Tỡm m phng trỡnh x(x 3)2 = m cú nghim phõn bit y x O -1 b) Ta cú: x(x 3)2 = m x3 6x2 + 9x = m Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Phng trỡnh cú ba nghim phõn bit v ch ng thng y = m ct (C) ti im phõn bit < m < < m < Cõu Cho hm s y = x x a) Kho sỏt s bin thiờn v v th ( C ) ca hm s ó cho b) Da vo th ( C ) hóy tỡm tt c cỏc giỏ tr ca tham s k phng trỡnh sau cú 2 bn nghim thc phõn bit x ( x ) = k + a v c PT honh giao im: x x = k + Lp lun c: S nghim PT ó cho chớnh l s giao im ca (C) v ng thng (d): y = k 4 + Lp lun c: YCBT < + Gii ỳng < k < Cõu Cho hm s y = k kx2+(3k-1)x+2k=0(x -1) < =>kx2+(3k-1)x+2k=0 ( vỡ x=-1 khụng phi l nghim ca pt vi mi k) k0 = k 6k + > Do ú d ct ( C ) ti im phõn bit k0 k < 2 (*) k > 2 Vy vi k thừa (*) thỡ thừa yờu cu bi toỏn Cõu Cho hm s y = - x3 + 3x2 - (C) a) Kho sỏt s bin thiờn v v th (C) ca hm s b) Tỡm m phng trỡnh x3 - 3x2 + m = cú nghim phõn bit th : Cho x = -1 y = , ( -1 ; ) Tõm i xng I (1;1) Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Tỡm m phng trỡnh x3 - 3x2 + m = cú nghim phõn bit Ta cú x3 - 3x2 + m = x3 - 3x2 = - m 3 (*) - x + 3x = m - x + 3x - = m - S nghim ca phng trỡnh (*) bng s giao im ca (C) v d: y = m Da vo th (*) cú nghim phõn bit - < m - < < m < Cõu Cho hm s y = 2x (C) x +1 1* Kho sỏt s bin thiờn v v th (C) 2* Tỡm m ng thng d: y = 2x + m ct th (C) ti im phõn bit A, B Phng trỡnh honh giao im: 2x2 + mx + m + = , (x - 1) d ct (C) ti im phõn bit PT(1) cú nghim phõn bit khỏc -1 m2 - 8m - 16 > m > + m < Cõu Cho hm s y = x x a*) Kho sỏt s bin thiờn v v th ca hm s b*) Tỡm m phng trỡnh x x = m + cú nghim phõn bit th (C) ca hm s nhn Oy lm trc i xng, giao vi Ox ti im ( ; 0) y 1O x b) Ta cú x x = m + x x = m (1) S nghim ca phng trỡnh (1) bng s giao im ca (C) v ng thng y = m Theo th ta thy ng thng y = m ct (C) ti im phõn bit v ch < m < Vy phng trỡnh ó cho cú nghim phõn bit m (4;3) Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Cõu Cho hm s y = x + 3x + cú th (C) a) Kho sỏt s bin thiờn v v th hm s (C) b) Da vo th (C) tỡm m phng trỡnh x 3x + m = cú nghim phõn bit th: im c bit: (0; 1), (-1; 3), (1; 3) x 3x + m = x + x + = m + S nghim ca phng trỡnh l s giao im ca th (C) vi ng thng y=m+1 Da vo th, phng trỡnh cú nghim phõn bit < m + < Cõu Cho hm s y = 13 0 ùù - ( m - 1) + m - ợ m - 6m + > m Vy giỏ tr m cn tỡm l m Cõu 12 Cho hm s y = mx3 - x - x + 8m cú th l ( Cm ) Tỡm m th ( Cm ) ct trc honh ti im phõn bit Phng trỡnh honh giao im: mx3 - x - x + 8m = (1) ự= ( x + 2) ộ ờmx - (2m +1) x + 4mỷ ỳ ộx =- ờmx - (2m +1) x + 4m = (2) ( Cm ) ct trc honh ti im phõn bit (1) cú ba nghim phõn bit (2) cú hai nghim phõn bit khỏc - Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com ùỡù m ù ùớ D =- 12m + 4m +1 > ùù ùùợ 12m + ỡù ùù ùù m ùỡù m ùù 1 ù - ùù ùớ P = m > ùù ùù S = 3m + > ợ ỡù ùù m ỡù ùù ùù m >ù ớmạ ùù ùù m ùợ ùù ùù m >3 ùợ ỡù ùù m >5 Vy giỏ tr m cn tỡm l ớù ùù m ợ mx - Cõu 14 Cho hm s y = cú th l ( Cm ) Tỡm m ng thng (d): x +2 y = x - ct th ( Cm ) ti hai im phõn bit A, B cho AB = 10 Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com Phng trỡnh honh giao im: mx - = 2x - x +2 (1) iu kin: x - Khi ú: (1) mx - = (2 x - 1)( x + 2) x - (m - 3) x - = (2) (d) ct ( Cm ) ti hai im phõn bit A, B (1) cú hai nghim phõn bit (2) cú hai nghim phõn bit khỏc - ỡù ự ùớ D = ộ ở- ( m - 3) ỷ + > ùùợ + 2m - - mạ - (*) t A( x1 ; x1 - 1) ; B ( x2 ; x2 - 1) vi x1 , x2 l hai nghim ca phng trỡnh (2) Theo nh lý Viet ta cú: Khi ú: m- ùỡù ùù x1 + x2 = ùù ùù x1 x2 =2 ùợ 2 ộ( x + x ) - x x ự= 10 AB = ( x1 - x2 ) + ( x1 - x2 ) = 10 2ỳ ở1 ỷ ổm ỗ ỗ ỗ ố 2 3ữ +2 = ữ ữ ứ m = [tha (*)] Vy giỏ tr m cn tỡm l m = Cõu 15 Cho hm s y = x - 3x + (m - 1) x +1 cú th l ( Cm ) Tỡm m th ( Cm ) ct ng thng (d ) : y = x +1 ti ba im A( 0;1) , B, C cho BC = 10 Phng trỡnh honh giao im: x3 - x + (m - 1) x +1 = x +1 (1) x ( x - x + m - 2) = ộx = ờx - x + m - = (2) ( Cm ) ct trc honh ti im phõn bit (1) cú ba nghim phõn bit (2) cú hai nghim phõn bit khỏc Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com ỡù 17 ỡùù D = - 4(m - 2) > ùù m < ùùợ m - ùù ùợ m (*) t B ( x1 ; x1 +1) ;C ( x2 ; x2 +1) vi x1 , x2 l hai nghim ca phng trỡnh (2) Theo nh lý Viet ta cú: Khi ú: ỡùù x1 + x2 = ùùợ x1 x2 = m - 2 2ộ = 10 (ởx1 + x2 ) - x1 x2 ự ỳ ỷ BC = ( x1 - x2 ) +( x1 - x2 ) = 10 - ( m - 2) = m =3 [tha (*)] Vy giỏ tr m cn tỡm l m = Cõu 16 Cho hm s y = x - (3m + 4) x + m cú th l ( Cm ) Tỡm m th ( Cm ) ct trc honh ti bn im phõn bit cú honh lp thnh mt cp s cng Phng trỡnh honh giao im: x - (3m + 4) x + m = (1) t t = x ( t 0) , phng trỡnh (1) tr thnh: t - (3m + 4)t + m = (2) (C) ct trc honh ti bn im phõn bit (1) cú bn nghim phõn bit (2) cú hai nghim dng phõn bit ỡù ùù m ỡù D = 5m + 24m +16 > ỡù ùù ùù ùù m >ù ù P =m >0 ớmạ ùù ùù ùù ùợ m ùù ùù S = 3m + > ợ ùù m >3 ùợ (*) Khi ú phng trỡnh (2) cú hai nghim < t < t2 Suy phng trỡnh (1) cú bn nghim phõn bit l x1 =- t2 < x2 =- t1 < x3 = t1 < x4 = t2 Bn nghim x1 , x2 , x3 , x4 lp thnh cp s cng x2 - x1 = x3 - x2 = x4 - x3 - ỡù t1 + t2 = 3m + Theo nh lý Viet ta cú: ùớù ùợ t1t2 = m Nguyn Vn Lc t1 + t2 = t1 t2 = t1 t2 = 9t1 (3) (4) (5) Ninh Kiu Cn Th 0933.168.309 www.TOANTUYENSINH.com 3m + ùỡù ùù t1 = 10 T (3) v (4) ta suy c ớù ùù t = 9(3m + 4) ùùợ 10 (6) Thay (6) vo (5) ta c: ộ3( 3m + 4) = 10m ( 3m + 4) = m ờ3 3m + =- 10m 100 ) ở( ộm =12 ờ 12 ờm =ờ 19 [tha (*)] ộm = 12 Vy giỏ tr m cn tỡm l 12 ờm =ờ 19 Nguyn Vn Lc Ninh Kiu Cn Th 0933.168.309 [...]... / x + 1 = k coự nghieọm 2x + 1 ữ x + 1 1 2x + 1 = k x + 2 (1) 3 = k (2) ( 2x + 1) 2 Th (2) vo (1) ta cú pt honh tip im l 1 3 x + ữ x + 1 2 = 2 2x + 1 ( 2x + 1) 1 1 3 (x 1) (2x + 1) = 3(x + ) v x x 1 = 2 2 2 5 1 x= Do ú k = 12 2 Vy phng trỡnh tip tuyn cn tỡm l: y = 1 1 x+ ữ 12 2 2x + 4 (C ) Cho hai im A (1; 0) v B(7; 4) Vit phng x +1 trỡnh tip tuyn ca (C ) , bit tip tuyn... 0933 .16 8.309 www.TOANTUYENSINH.com 1. 4.4 Tip tuyn song song vi ng thng d x 1 Vit phng trỡnh tip tuyn ca th hm s bit tip x +1 x 2 tuyn song song vi d: y = 2 2 x 1 ( x 1) y= y = x +1 ( x + 1) 2 Cõu 1 Cho hm s y = x 2 1 1 cú h s gúc k = TT cú h s gúc k = 2 2 2 1 2 1 x = 1 = 0 Gi ( x0 ; y0 ) l to ca tip im Ta cú y ( x0 ) = 2 2 2 ( x0 + 1) x 0 = 3 1 1 + Vi x0 = 1 y0 = 0 PTTT: y = x 2 2 1 7... m + 2 x1 x2 = m 1 Khi ú A ( x1 ; x1 m ) , B ( x2 ; x2 m ) Theo h thc Viet ta cú AB = 3 2 AB 2 = 18 2 ( x1 x2 ) = 18 ( x1 x2 ) = 9 2 2 ( x1 + x2 ) 4 x1 x2 = 9 ( m + 2 ) 4 ( m 1) = 9 m = 1 2 2 Cõu 4 Cho hm s y = 2x + 1 x 1 Tỡm im M trờn (C) khong cỏch t M n tim cn ng ca th (C) bng khong cỏch t M n trc Ox Gi M ( x 0 ; y 0 ) , ( x 0 1) , y0 = x0 1 = 2x 0 + 1 , Ta cú d ( M, 1 ) = d... tuyn vi (C) ti hai Nguyn Vn Lc Ninh Kiu Cn Th 0933 .16 8.309 www.TOANTUYENSINH.com im ú vuụng gúc vi nhau Gi s trờn (C) cú hai im A( x1; y1 ), B( x2 ; y2 ) vi x1, x2 > 3 sao cho tip tuyn vi (C) ti hai im ny vuụng gúc vi nhau Khi ú, ta cú: y '( x1 ) y '( x2 ) = 1 (3x12 12 x1 + 9)(3x22 12 x2 + 9) = 1 9 ( x1 1) ( x1 3) ( x2 1) ( x2 3) = 1 (*) Do x1 > 3 v x2 > 3 nờn VT(*) > 0 Do ú (*) vụ lớ Vy: Trờn... 1 = 2x 0 + 1 , Ta cú d ( M, 1 ) = d ( M, Ox ) x 0 1 = y 0 x0 1 2x 0 + 1 2 ( x 0 1) = 2x 0 + 1 x0 1 x0 = 0 1 2 , ta cú : x 0 2x 0 + 1 = 2x 0 + 1 Suy ra M ( 0; 1) , M ( 4;3) 2 x0 = 4 1 Vi x 0 < , ta cú pt x 20 2x 0 + 1 = 2x 0 1 x 02 + 2 = 0 (vụ nghim) 2 Vy M ( 0; 1) , M ( 4;3 ) Vi x 0 x (1) x 1 a) Kho sỏt s bin thiờn v v th (C) ca hm s (1) b) Tỡm m ng thng y = x + m ct th (C) ti hai im... 6 x0 = 3 x02 2 x0 + 1 = 0 x0 = 1 Vỡ x0 = 1 y0 = 2 M (1; 2) Phng trỡnh tip tuyn cn tỡm l y = 3( x 1) 2 y = 3 x + 1 Cõu 3 : Cho hm s: y = 2 x 3 7 x + 1 (C) Vit phng trỡnh tip tuyn ca th (C) cú h s gúc k = 1 y = 2 x3 7x + 1 y ' = 6 x2 7 x0 = 1 x0 = 1 2 Gi ( x0 ; y0 ) l to ca tip im Ta cú: y ( x0 ) = 1 6 x0 7 = 1 Vi x0 = 1 y0 = 6 PTTT : y = x + 7 Vi x0 = 1 y0 = 4 PTTT : y =... v d: y=-x+m l: x 1 x 1 x+2 = x + m 2 2 x 1 x + 2 = x + mx + x m x mx + m + 2 = 0 (1) d ct (C) ti hai im phõn bit khi v ch khi (1) cú hai nghim phõn bit khỏc 1 1 m + m + 2 0 2 m 2 4m 8 > 0(*) m 4(m + 2) > 0 Khi ú d ct (C) ti A( x1; x1 + m), B ( x2 ; x2 + m) , vi x1 , x2 l nghim phng trỡnh (1) Theo Viet, ta cú AB = ( x2 x1 ) 2 + ( x1 x2 ) = 2 ( x1 + x2 )2 4 x1.x2 = 2 ( m 2 4m... - cos x +1 Tp xỏc nh: D = Ă 2 ự t t = cos x vi t ẻ ộ ở- 1; 1ỷ, hm s tr thnh: y =- 2t - t + 3 Ta cú: y ' =- 4t - 1 ; y ' = 0 t =- 1 ộ ự ẻ - 1; 1 4 ở ỷ ổ 1 25 25 ữ ữ = ị min y = 0; max y = ữ ữ x ẻ D x ẻ D ố 4ứ 8 8 ỗDo y ( - 1) = 2; y ( 1) = 0; y ỗ ỗ y =- 2 2; max y = 2 Vy min xẻ D xẻ D Nguyn Vn Lc Ninh Kiu Cn Th 0933 .16 8.309 www.TOANTUYENSINH.com 1. 4 Tip tuyn 1. 4 .1 Tip tuyn ti mt im Cõu 1 Vit phng... cú th (H) Vit phng trỡnh tip tuyn ca (H) bit x 1 1 tip tuyn song song vi ng thng y = x + 5 8 2 x +1 y= y = x 1 ( x 1) 2 1 Vỡ tip tuyn song song vi ng thng y = x + 5 nờn h s gúc ca tip tuyn l 8 1 k= 8 ( x0 ; y0 ) Gi l to ca tip im Cõu 5: Cho hm s y = y ( x 0 ) = k x = 3 1 = ( x0 1) 2 = 16 0 8 ( x0 1) 2 x0 = 5 2 1 2 1 1 x + 3) + ( 8 2 3 1 3 Vi x0 = 5 y0 = PTTT : y = ( x 5 ) + 2 8... 12 = 0 (tha món (*)) m = 6 Vy m = 2 hoc m = 6 Nguyn Vn Lc Ninh Kiu Cn Th 0933 .16 8.309 www.TOANTUYENSINH.com Cõu 3 Tỡm m ng thng ( d ) : y = x m ct th ( C ) ca hm s y = x +1 ti hai x 1 im A, B sao cho AB = 3 2 x +1 = x m x + 1 = ( x m ) ( x 1) (vỡ x = 1 khụng l nghim ca x 1 2 pt) x ( m + 2 ) x + m 1 = 0 (1) Pt honh giao im Pt (1) cú 2 nghim phõn bit x1 , x2 = m 2 + 8 > 0 m Ă x1