Chuyên đề hàm số hay

Gia s Thnh c www.daythem.edu.vn o hm VD Tỡm o hm ca cỏc hm s sau : 1, y 2x4 x3 x 3, y x 2 x x 3 2, y ( x3 2)(1 x2 ) x 4, y x 4x 6, y 5, y x5 x3 x x ; x 3 x x x b a2 c x b ( a , b , c l hng s) a x VD Tỡm o hm ca cỏc hm s sau : 1, y 2x 1 3x 2x 2x2 x 7, y 10, y x x x ( x x 1) x 12, y 2, y 2x VD Tỡm o hm ca cỏc hm s sau : 4, y x x 9, y 11, y x 1, y (2 x3 3x x 1) x x2 x x2 6, y x x 2x 8, y x 10, y 3, y 5, y x(2 x 1)(3x 2) 4, y (2 x 3)( x5 x) 7, y x 3x x 2, y 5x x x 3, y ( x2 x 1)3 ( x2 x 1)2 5, y x x x x x 8, y x x ( x 1)2 6, y x x ; 9, y ( x2 x 1)4 12, y 2x 11, y ( x 2) x2 ( x 1)3 VD Tỡm o hm ca cỏc hm s sau : 2, y 1, y 2sin 3x cos x 4, y (sin x cos x) sin x cos x 5, y tan x cot x 2 sin x cos x 6, y tan2x tan3 2x tan5 2x 3, y tan x tan x 3, y sin x cos x sin x cos x ; 7, y tan sin cos3 x VD Tỡm o hm ca cỏc hm s sau : sin x x sin x cos x 1, y 2, y sin x cos x x sin x sin x cos x 4, y sin x cos x.sin x 5, y 4 7, y cos x sin x 8, y (sin x cos x) sin x cos x 6, y sin x x cos x cos x x sin x 9, y sin x cos3 x Gia s Thnh c www.daythem.edu.vn 10, y sin cos3 x VD Tỡm o hm ca cỏc hm s sau : 1, y sin x cos x sin x cos6 x 6 x3 x2 12, y cot cos2 2, y cos x 2cos x sin x 2sin x sin x 3cos x 4, y sin x cos x 3cos x x tan sin x 6, y sin x 8, y 2cos x , x ; 3, y sin x cos x cos x 2sin x 6sin x 11, y sin cos cos3 x x cos x sin x sin x sin x sin x 7, y cos x cos x cos3 x cos x VD Cho hm s y x sin x chng minh : 5, y cos x cos y' x tan x cos x VD Cho cỏc hm s : f x sin x cos4 x , g x sin x cos6 x Chng minh : f ' x g ' x 1, xy y ' sin x x 2cos x y 2, VD 10 1, Cho hm s y x x Chng minh : x y' y 2, Cho hm s y cot x Chng minh : y ' y VD 11 Gii phng trỡnh y ' bit : 2, y cos x sin x 1, y sin x cos x 3, y 3sin x cos x 10 x f x x / x mx Tỡm m : VD 12 Cho hm s : 1, f x x 2, f x , x 0; ; 3, f x , x 0; 4, f x , x ; VD 13 Cho hm s : f x 1, f x , x ; m m x x m x 5m Tỡm m : 2, f x cú hai nghim cựng du VD 14 Cho hm s y x3 2m x mx Tỡm m : 1, y ' cú hai nghim phõn bit 3, y ' , x ; 2, y ' , x 4, y ' , x VD 15 Cho hm s y mx3 m x mx Xỏc nh m : 1, y ' , x 2, y ' cú hai nghim phõn bit cựng õm 3, y ' cú hai nghim phõn bit tha iu kin : x12 x22 BàI: PHƯƠNG TRìNH TIếP TUYếN Gia s Thnh c www.daythem.edu.vn 1, Khi bit tip im : Tip tuyn ca th C : y f x ti M x0 ; y0 , cú phng trỡnh l : y f ' x0 x x0 y0 ( ) 2, Khi bit h s gúc ca tip tuyn: Nu tip tuyn ca th C : y f x cú h s gúc l k thỡ ta gi M x0 ; y0 l tip im f ' x0 k (1) Gii phng trỡnh (1) tỡm x0 suy y0 f x0 Phng trỡnh tip tuyn phi tỡm cú dng : y k x x0 y0 Chỳ ý : Hai ng thng song song vi thỡ h s gúc ca chỳng bng Hai ng thng vuụng gúc nu tớch h s gúc ca chỳng bng 3, Bit tip tuyn i qua im A x1 ; y1 VD Cho ng cong C : y f x x3 3x Vit phng trỡnh tip tuyn ca C 1, Ti im M ; 3, Ti giao im ca C vi trc honh VD 2.Cho ng cong C : y 2, Ti im thuc C v cú honh x0 4, Bit tip tuyn i qua im A ; 3x 1 x 1, Vit PTTT ca C bit tip tuyn song song vi d : x y 21 2, Vit PTTT ca C bit tip tuyn vuụng gúc vi : x y VD 1, Cho hm s y x3 3x x C Tỡm tip tuyn cú h s gúc nh nht 2, Cho hm s y x3 3x x C Tỡm tip tuyn cú h s gúc ln nht VD Vit phng trỡnh tip tuyn ca th hm s y x2 2x 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 VD 5.Cho C l th ca hm s y x x CMR tip tuyn ti mt im bt kỡ ca C ct trc tung ti mt im cỏch u gc ta v tip im VD 6.Cho hm s C : y x3 x Vit phng trỡnh tip vi C : 1, Ti im cú honh x0 2, Bit tip tuyn song song vi: x y ; VD 7.Cho hm s : y 3x 1 x C 1, Vit PTTT ca C ti im M ; 2, Vit PTTT ca C ti giao ca C vi Ox 3, Vit PTTT ca C ti giao ca C vi Oy 4, Vit PTTT ca C bt TT // d : x y 5, Vit PTTT ca C bit tip tuyn vuụng gúc vi : x y VD 9.Cho hm s y x x C Tỡm phng trỡnh tip tuyn vi C : VD 10.Cho hm s y x 3mx m x 2, Song song vi: d : x y 1, Ti im cú honh x0 Tỡm cỏc giỏ tr ca hm s (1) ti im cú honh x i qua im A ;2 VD 11.Cho hm s y 3x x m tip tuyn ca th ca Tớnh din tớch ca tam giỏc to bi cỏc trc ta v tip tuyn ca Gia s Thnh c www.daythem.edu.vn th ca hm s (1) ti im M ; VD 12.Cho hm s y 3x3 C Vit phng trỡnh tip tuyn ca th C bit tip tuyn to vi ng thng d : y x gúc 300 VD 13.Cho hm s y 2x x C Gi I ; Tỡm im M C cho tip tuyn ca C ti M vuụng gúc vi ng thng IM 2x VD 14 Cho hm s y C Tỡm im M C , bit tip tuyn ca C ti M ct hai trc ta x ti A , B v tam giỏc OAB cú din tớch bng (Khi D - 2007) x C Vit phng trỡnh tip tuyn ca C cho v hai x ng d1 : x ; d : y ct to thnh mt tam giỏc cõn VD 15 Cho hm s : y 2x Vit PTTT vi th (C) bit h s gúc ca tip tuyn bng - x2 3x VD 17 Cho hm s y (C) Vit PTTT vi (C) bit tip tuyn ú vuụng gúc vi y x 10 x 3x VD 18.Cho hm s y (C ) Vit phng trỡnh tip tuyn vi th (C) i qua im A (2; 0) x VD 19 Cho hm s y x3 3x (C) Vit PTTT ca (C) bit h s gúc bng (TN THPT 2013) x VD 20.Cho hm s y (C) Vit PTTT ca (C) bit tip tuyn i qua im P(3;1) x VD 16.Cho hm s y VD 21 Cho hm s y x 3x (C) 1, Vit PTTT (C) ti im M 2;4 2, Vit PTTT (C) ti im cú honh x 3, Vit PTTT ca (C) ti cỏc im cú tung y VD 22 Cho hm s y = - 2x + 3x - (C) 1, Vit phng trỡnh tip tuyn ca (C), bit tip tuyn vuụng gúc vi d : y x 2015 2, Vit phng trỡnh ng thng i qua M 1; v tip xỳc vi th (C) 4 VD 23 Cho hm s y x x (C) 1, Vit phng trỡnh tip tuyn ca th (C) ti im cú honh x 2, Vit phng trỡnh tip tuyn ca th (C) ti im cú tung y 3,Vit phng trỡnh tip tuyn ca th (C) , bit h s gúc ca tip tuyn bng 24 Gia s Thnh c www.daythem.edu.vn Bài: Hàm số đồng biến, NGHịCH BIếN Bi toỏn: Xột s bin thiờn ca hm s y = f(x) P2: Ta cn thc hin cỏc bc sau: B1: Tỡm xỏc nh ca hm s B2: Tớnh o hm f (x), ri gii phng trỡnh f (x) = B3: Lp bng bin thiờn ca hm s B4: Kt lun VD Tỡm cỏc khong ng bin, nghch bin ca hm s 1, y = 3x2 8x3 2, y = x3 6x2 + 9x 4, y x3 x x 5, y = x2(4 x2) 8, y x x x 7, y x3 x2 17 x 3, y x3 x 6, y = x4 + 8x2 + 9, y x3 x VD Lp bng bin thiờn v tỡm cỏc khong ng bin, nghch bin ca hm s x2 x 2, y x 2x 1, y x7 4, y x 4x 5, y 7, y x2 2x 2x x2 x 8, y x x2 3, y x2 x x 6, y x2 x3 9, y x 2x x2 Bi toỏn: Xỏc nh m hm s y = f(x, m) ng bin (hay nghch bin) trờn khong D B1: Tỡm xỏc nh ca hm s B2: Tớnh o hm f(x) B3: Lp lun cho cỏc trng hp f (x) vi x D f '(x) xD f (x) vi x D max f '(x) xD VD Tỡm m sau cho hm s: 1, y = x3 3(m 1)x2 + 3m(m-2)x + B / R 3, y = x3 + 3x2 + (m + 1)x + 4m NB/ 1;1 5, y = 2, y = mx3 (2m 1)x2 + (m 2)x B / R 4, y = (m2 + 5m)x3 + 6mx2 + 6x n iu / R m x + mx2 + (3m 2)x B / R 3 6, y = - x3 + (m 1)x2 + (m + 3)x B /(0; 3) 7, y = x3 3(2m + 1)x2 + (12m + 5)x + B /(2; + ) 8, y = x3 (m+1)x2 (2m2 3m + 2)x + 2m(2m 1) ng bin x 9, y = x3 3mx2 + m ng bin khong (- ; 0) VD Tỡm m sau cho hm s: 1, y = (2m + 3)sin2x + (2 m)x B / R 2, y = 2x + mcosx, tng trờn R 3, y = x + msinx, ng bin trờn R 4, y = (m 3)x (2m + 1)cosx, nghch bin trờn R VD Tỡm m sau cho hm s: x 2( m 1) B/ (0; + ) x x 3x m y 4, B / (3; + ) x mx luụn nghch bin x m3 x x m y 3, NB/ ( ; ) 2x 2, y 1, y Gia s Thnh c www.daythem.edu.vn Bài: cực trị Hàm số VD Tỡm cc tr ca hm s 3 1, y x3 x 3x 2, y x3 x 3x 3, y x3 x 12 x 4, y x3 3x x 5, y 3x x3 24 x 48 x 6, y x 2, y x x 3, y x x ; x2 VD Tỡm cc tr ca hm s 1, y x ; x VD Tỡm a , b , c hm s y x3 ax bx c t cc tiu ti x , y v th ca hm s ct trc tung ti im cú tung bng VD Cho hm s y x3 3x Vi giỏ tr no ca m ng thng ni hai im cc tr ca hm s tip xỳc vi ng trũn (C) : x m y m VD Cho hm s y = VD x + ( m )x2 + (2m )x (1) Tỡm m hm s (1) cú hai cc tr Tỡm m hm s y m x3 3x mx cú cc i, cc tiu VD Cho hm s y = x3 + ( 2m )x2 + 3x + m (1) Tỡm m hm s ( 1) khụng cú cc tr VD Cho hm s: y m x mx Vi giỏ tr no ca m thỡ th ca hm s khụng cú im cc i v im cc tiu 2 VD Cho hm s: y x mx m m x Tỡm m hm s t cc tiu ti im x=1 VD 10 Cho hm s y = x + mx2 + (m2 )x +2 (1) Tỡm m hm s ( ) t cc i ti x = VD 11 Cho hm s y x 3x 3x 1, Tỡm cc tr ca hm s 2, Vit phng trỡnh ng thng i qua cỏc im cc tr VD 12 Cho hm s y x x m x m Xỏc nh m cho: 1, Hm s cú cc tr 2, Hm s cú hai cc tr cựng du 3 VD 13 Tỡm m hm s y mx3 m x m x t cc i, cc tiu ti x1, x2 tho x1 x2 x3 x mx t cc i v cc tiu cú honh ln hn m VD 15 Cho hm s: y f x x3 m x m x (1) Tỡm m (1) cú cc i, cc tiu v ng thng i qua im cc i, cc tiu song song vi ng thng y 3x VD 14 Tỡm m hm s y x 3x VD 16 Cho hm s: y Vit phng trỡnh ng thng i qua cỏc im cc tr x2 x mx m VD 17 Cho hm s: y m Tỡm m cú giỏ tr cc i v giỏ tr cc tiu Vit xm phng trỡnh ng thng i qua cỏc im cc tr Gia s Thnh c www.daythem.edu.vn VD 19 Tỡm m hm s y = x - ( m + )x2 + (m2 + )x + m t cc tr ti x1 , x2 tha x12 + x22 = 10 1 x1 x2 VD 20 Tỡm m y = x3 3mx2 (2m+3)x + t cc tr ti x1 , x2 tha x1 x2 VD 21 Cho hm s y = x mx2 (3m2 )x + 3 (1) Tỡm m hm s (1) t cc tr ti x1, x2 tha x1x2 + 2(x1 + x2 ) = VD 22 Cho hm s y = 2x3 ( 9m + )x2 + 12m(m+1)x m3 Tỡm m hm s ( 1) t cc tr ti x1, x2 tha x1 x2 = VD 24 Cho hm s y = x3 (2m )x2 + (2 m )x + (1) Tỡm cỏc giỏ tr ca m hm s ( ) cú cc i, cc tiu v cỏc im cc tr ca th hm s ( 1) cú honh dng VD 25 Tỡm m 1, th hm s y = x3 (2m + )x2 + (m2 3m + )x + cú cc i, cc tiu v im cc i, cc tiu ca th hm s nm v hai phớa i vi trc tung 2, th hm s y x3 3x 6mx m cú hai im cc tr cựng mt phớa i vi trc honh VD 26 Cho hm s y = x3 3mx2 + 3m3 (1) Tỡm m (Cm) cú im cc tr A v B cho tam giỏc OAB cú din tớch bng 48 VD 27 Cho hm s y = -x3 + 3x2 + 3(m2 )x 3m2 ( 1) Tỡm m hm s ( ) cú cc i, cc tiu v cỏc im cc tr ca th hm s ( ) cỏch u gc ta O VD 28 Cho hm s y = 2x3 3(2m + 1)x2 + 6m(m+1)x + (1 ) Tỡm m hm s ( ) cú cc tr Khi ú chng minh rng khong cỏch gia hai im cc tr ú khụng i VD 29 Cho hm s y x3 3mx 3(m2 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 gúc ta O bng ln khong cỏch t im cc tiu ca th hm s n gc ta O x mx 4mx (1) Tỡm m hm s ( ) cú cc tr ti x1, x2 tha 2 x 5mx1 12m m biu thc A t giỏ tr nh nht x1 5mx2 12m m2 VD 30 Cho hm s y = VD 33 Tỡm m hm s y x3 3(m 1) x 6mx m3 cú cc i , cc tiu : yC yCT VD 34 Cho hm s y x3 (m 1) x m2 x x Vi giỏ tr no ca m hm s cú cc i cc tiu x1 , x2 Tỡm GTLN A = x1 x2 2( x1 x2 ) VD 36 Cho hm s y x3 mx m Tỡm m im cc i v im cc tiu ca d th hm s nm v hai phớa ca d : x y VD 37 Tỡm m y x3 2(m 1) x (m2 4m 1) x 2(m2 1) cú cc tr ti x1 , x2 sao: 1 x1 x2 x1 x2 Gia s Thnh c www.daythem.edu.vn VD 39 Cho hm s : y x3 3(m 1) x x m Tỡm m hm s t cc i v cc tiu ti cỏc im cú honh x1 ; x2 : x1 x2 VD 40 Cho hm s y x3 (1 2m) x (2 m) x m Tỡm m th hm s cú im cc i v cc tiu, ng thi honh ca cc tiu nh hn VD 41 [HB11] Cho hm s y x m x m Tỡm m th hm s cú ba im cc tr A , B , C cho OA BC ; ú A thuc trc tung, B v C l hai im cc tr cũn li VD 42 [HA12] Tỡm m th hm s y x m x m2 cú ba im cc tr to thnh ba nh ca mt tam giỏc vuụng VD 43 Cho hm s y x 2mx 2m m4 ( m l tham s) Tỡm m th hm s cú cc i v cc tiu lp thnh mt tam giỏc u VD 46 Tỡm m th hm s y x 2mx m cú im cc tr to thnh mt tam giỏc cú din tớch bng VD 47 Cho hm s y x 2mx m Tỡm m th hm s cú im cc tr to thnh mt tam giỏc cú bỏn kớnh ng trũn ngoi tip bng Bài: ứng dụng đạo hàm Phn 1: Giỏ tr ln nht, nh nht 1, N: Cho hm s xỏc nh trờn D f ( x) M , x D +Nu thỡ max f ( x) M xD xo D : f ( xo ) M Gia s Thnh c www.daythem.edu.vn f ( x) m, x D +Nu thỡ f ( x) m xD xo D : f ( xo ) m 2, PHNG PHP GII TON Bi toỏn Cho hm s y=f(x) liờn tc trờn khong (a;b) (a cú th l -, b cú th l +) Hóy tỡm max f ( x) ( a ;b ) v f ( x) (nu chỳng tn ti) ( a ;b ) Cỏch gii Lp bng bin thiờn Cho hm s y=f(x) liờn tc trờn on [a;b] Hóy tỡm max f ( x) v f ( x) Bi toỏn [ a ;b ] [ a ;b ] Cỏch gii 1, ỡ cỏc i ti h n 1, x2, ., n ca ( t n o n a;b] 2, ớnh (a , ( 1), f(x2 , , ( n), f(b) 3, max f ( x) max f a , f x1 , f x2 , , f b ; f ( x) f a , f x1 , f x2 , , f b [ a ;b ] [ a ;b ] 3, B I T P P D NG VD Tỡm GTLN-GTNN ca cỏc hm s sau: 1, y f x x3 x trờn on 1;1 2, y f x x x trờn on 0; 3, y f x x3 x x trờn on 1;0 VD Tỡm GTLN-GTNN ca cỏc hm s sau: 1, y f x x3 x trờn on 1;3 2, y f x x x 3, y f x x3 3x 12 x trờn on 2; 4, y f x x3 3x trờn on 1; 5, y f x x x 16 trờn on 1;3 6, y f x x x trờn on 0; trờn on 0; 2 VD Tỡm GTLN-GTNN ca cỏc hm s sau: 1, y f x 2x trờn on 2; x 2, y f x 2x trờn on x2 ;1 x2 2x 4, y f x trờn on 0;3 x2 3, y f x x trờn on 1; x2 VD Tỡm GTLN-GTNN ca cỏc hm s sau: 1, y x x2 2, y x x x x 1, x 1;1 3, y x x trờn on 0;3 4, 5, y x x2 6, y x x trờn on 0; trờn on 1; VD Tỡm GTLN-GTNN ca cỏc hm s sau: 1, y f x sin x x trờn on ; 2, y f x x cos x trờn on 0; 2 Gia s Thnh c www.daythem.edu.vn 3, y f x 2sin x sin x trờn on 0; 4, y f x cos x 4sinx trờn 0; 5, y f x 2sin x cos x 4sin x 6, y f x s inx trờn on 0; cos x VD Tỡm GTLN v GTNN ca hm s : 1 x2 x x x x x x 8x 8x 4, y x2 x 1, y x x ( x 1)(3 x) 3, y x2 x 5, y 2, y x3 x x4 x2 x2 x2 x2 VD Tỡm GTLN v GTNN ca hm s : sin x cos x sin x cos x sin x 3, y sin x sin x 1, y 2, y = sin4x +cos4x +sinx.cosx +1 4, y = sinx + cosx + sinx cosx 6, y 5, y 2cos x 2sin x 7, y cos4 x sin x sin x cos2 x cos x cos x cos x 1 8, y 2(1 sin x cos x) (cos x cos8 x) Phn 2: Giỏ tr ln nht, nh nht ca biu thc 1, Phng phỏp gii tỡm GTLN, GTNN ca mt biu thc cú cha nhiu hn mt bin s no ú ta cú th dựng phng phỏp i bin s nh sau: Bc Biu din cỏc bin s ca biu thc ban u theo mt bin s mi 10 Gia s Thnh c www.daythem.edu.vn Bc Tỡm iu kin cho bin s mi (da trờn iu kin ca cỏc bin s ban u) Bc Tỡm GTNN, GTLN ca hm s theo bin s mi tng ng vi iu kin ca nú 2, Mt s bt ng thc c s thng s dng: a, Vi a, b, c bt k, ta cú: 1, a b 2ab 2, (a b) 4ab 3, 2(a b ) (a b) 4, a b c ab bc ca 5, (a b c) 3( ab bc ca) 6,3(a b c ) ( a b c) b, BT Cụsi - Vi a, b, c khụng õm, ta cú: a b ab , a b c 3 abc , a b c 27abc ab a2 b2 ab 1 11 a b ab a b ab a b a b a3 b3 2 (a 0, b 0) (a 0, b 0) (a 0, b 0) (a c) (b d ) a b c d 2 1 a b ab a b ab (a, b R) (0 ab 1) (ab 1) 3, Tỡm GTLN, NN ca biu thc M cú tớnh cht sau: Tớnh cht 1: M ph thuc vo i lng: x + y + z, xy + yz + zx hoc x2 + y2 + z2 Tớnh cht 2: Gi thit cho trc giỏ tr ca mt i lng: x + y + z, xy + yz + zx hoc x2 + y2 + z2 Cỏch gii: Gi s biu thc M cú mt i lng nờu trờn, ú cú th t mt hai i lng ca biu thc M l n ph t ri dựng gi thit ca bi toỏn ó cho v kt hp hng ng thc (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx biu din i lng cũn li theo t Tỡm K cho t ta thng dựng mt ba BT sau: x2 + y2 + z2 xy + yz + zx hoc (x + y + z)2 3(xy + yz + zx) hoc 3(x2 + y2 + z2) (x + y + z)2 Quy v bi toỏn n gin a b4 a b2 a b VD Tỡm giỏ tr nh nht ca F Vi a,b b a b a b a VD Cho x, y l hai s thc dng tha x y Tỡm GTNN ca biu thc P x 4y VD Cho hai s thc x, y thoó món: x 0, y , x+y = Tỡm giỏ tr nh nht, ln nht ca biu thc : P = x3 + 2y2 +3x2 + 4xy - 5x 11 Gia s Thnh c www.daythem.edu.vn VD Cho x, y, z l cỏc s dng Tỡm GTNN ca biu thc P xyz x yz xyz x yz VD Tỡm GTLN, NN ca A x y Bit x, y tho iu kin x y x y VD Tỡm GTNN ca Q xy VD Cho x, y tha x2 + y2 = 2, Tỡm GTLN, NN ca M = (x3 + y3) 3xy x y 1 vi x, y dng v x khỏc y x2 y x, y Tỡm GTLN, GTNN P ( x 1)( y 1) x y x y VD Cho x, y tha x, y x y Tỡm GTLN, GTNN ca P x y xy x y xy Cho x, y tha VD 10 VD 11 Cho x, y l hai s thc khụng õm tha x3 y xy Tỡm GTLN, GTNN ca biu thc P x y xy x, y x2 y2 VD 12 Cho x, y tha Tỡm GTLN, GTNN P y x x y x y xy x y VD 13 Cho cỏc s thc dng tho: x + y = Tỡm GTNN ca: P x y VD 15 Cho y 0, x x y 12 Tỡm GTLN, NN ca: P=xy + x + 2y +17 Cho x, y l cỏc s thc khụng õm thay i v tha iu kin 4( x y xy ) 2( x y ) Tỡm giỏ tr ln nht ca biu thc P xy x y x y VD 22 Cho cỏc s thc khụng õm x, y, z tho x y z VD 21 2 Tỡm giỏ tr ln nht ca biu thc P xy yz zx x yz VD 23 Cho x, y , z tha x2 + y2 + z2 = Tỡm GTLN, NN ca R = x3 + y3 + z3 3xyz VD 24 Cho x, y , z > v x + y + z Tỡm GTNN ca M = x + y + z x 1 y z VD 25 Cho cỏc s x, y , z thuc khong (0 ; 1) v tha xyz + (x 1)(y 1)(z 1)=0 Tỡm GTNN ca N = x2 + y2 + z2 VD 26 Cho x, y, z 0, x y z Chng minh: x2 1 y z 82 (H- A-03) x y z VD 28 Cho x,y,z l cỏc s thc khụng õm tha x+y+z=1 Tỡm GTLN ca: P=xy+yz+zx-2xyz VD 29 Cho x,y,z l cỏc s thc khụng õm tha x+y+z=1 Tỡm GTNN ca P x3 y z VD 30 Cho x, y, z 1; Tỡm GTLN ca biu thc P ( x y z ) x 12 1 y z 15 xyz Gia s Thnh c www.daythem.edu.vn VD 31 Cho x, y, z v x y z Tỡm GTLN ca P x y z VD 32 Cho x y z Tỡm GTLN v GTNN ca biu thc P x y z xy yz zx VD 33 Cho x,y,z l cỏc s thc khụng õm tha x+y+z=1 VD 34 27 Cho x, y, z v x y z Tỡm GTLN ca P x3 y z VD 35 Cho x, y, z v x y z Tỡm GTLN ca P xy 10 xz 22 yz VD 36 Cho ba s thc dng a, b, c tha a b c Tỡm GTNN ca P a b3 c3 Chng minh: xy yz zx xyz Phn 3: Chng minh bt ng thc bng phng phỏp tip tuyn Xột bi toỏn: Cho a1 , a2 , a3 , , an D tho a1 a2 a3 an n , vi D , cn chng minh bt ng thc f a1 f a2 f an nf , ng thc xy a1 a2 a3 an VD Cho bn s dng a, b, c, d tho a b c d CMR: a b3 c3 d a b c d VD Cho ba s thc dng a, b, c tho a b c CMR: 10 a3 b3 c3 a5 b5 c5 VD 2a b c 2b c a 2c a b Cho cỏc s thc dng a, b, c CMR: 2 2 2a b c 2b c a 2c a b VD Cho ba s thc dng a, b, c tho a b c CMR: 13 2 1 27 ab bc ca 8 Gia s Thnh c www.daythem.edu.vn VD Cho ba s thc dng x , y , z tha x y z 12 CMR: VD Cho bn s thc dng a, b, c, d tha a + b + c + d = , CMR: a + 3a VD + b + 3b2 + c + 3c + d + 3d Ê 1 1 x y z Cho ba s thc a, b, c tha a b c CMR: a b2 c 10 a b c a c b c b a l cỏc s thc dng CMR: 2 2 c b a b2 a c a b c VD 10 Cho a,b,c Bài: khảo sát hàm số 1, Kho sỏt v v th hm s *) TX: D *) SBT: CBT: Tớnh y Du y v suy CBT 14 2 Gia s Thnh c www.daythem.edu.vn Cc tr Gii hn Bng bin thiờn, *) V th 2, Bin lun s nghim phng trỡnh da vo th (C ): y=f(x) - a phng trỡnh v dng f(x) = A(m) - S nghim ca phng trỡnh l s giao im ca th hm s y = f(x) vi ng thng y = A(m) - V hai th lờn cựng mt h trc ta v bin lun kt qu VD Cho hm s y x3 3x 1, Kho sỏt v v th hm s trờn 2, Da vo th bin lun theo m s nghim ca phng trỡnh x3 3x2 m VD Cho hm s y x3 x 1, Kho sỏt s bin thiờn v v th (C) ca hm s 2, Vit phng trỡnh tip tuyn ca (C) ti im trờn (C) cú tung bng 3, Vit phng trỡnh tip tuyn ca (C) ti im cú honh bng VD Cho hm s y x3 3(m2 1) x 6mx 2m 1, Kho sỏt s bin thiờn v v th (C) ca hm s m = 2, Tỡm giỏ tr ca m hm s t cc tr ti x = Khi ú xỏc nh giỏ tr cc tr ca hm s ti ú VD Cho hm s y x3 3x cú th (C) 1, Kho sỏt s bin thiờn v v th (C) 2, Vit phng trỡnh tip tuyn vi (C) bit tip tuyn song song vi d : y x 3, Tỡm GTLN, GTNN ca hm s y x3 3x trờn [1; 3] VD Cho hm s y x3 mx m , m l tham s 1, Kho sỏt v v th hm s (C) ca hm s m = 3 2, Vit phng trỡnh tip tuyn ca (C) bit tip tuyn vuụng gúc vi d : y x 3, Xỏc nh m hm s t cc tiu ti im x = VD Cho hm s y x x cú th (C ) 1, Kho sỏt s bin thiờn v v th (C ) 2, Vit phng trỡnh tip tuyn ca th (C) ti im cú honh x0 = VD Cho hm s y x 3x cú th (C) 1, Kho sỏt s bin thiờn v v th hm s (C) 2, Da vo th (C) tỡm m phng trỡnh x 3x m cú nghim phõn bit 3, Tỡm GTLN, GTNN ca hm s y x 3x trờn [0; 2] VD Cho hm s y x mx (m 1) cú th (Cm) 1, Kho sỏt s bin thiờn v v th (C) ca hm s m = -2 2, Tỡm m th hm s i qua im M(-1;4) 3, Tỡm m hm s y x mx (m 1) cú cc i v cc tiu VD 10 Cho hm s y x 3x cú th (C) 1, Kho sỏt s bin thiờn v v th (C) ca hm s 2, Vit phng trỡnh tip tuyn vi (C) ti im cú honh x0 = 15 Gia s Thnh c www.daythem.edu.vn 3, Tỡm m phng trỡnh sau cú nghim phõn bit: x4 12 x2 m VD 11 Cho hm s y x m x (Cm ) 2 1, Kho sỏt hm s m = 2, Bin lun theo m s nghim ca phng trỡnh x x 4m 3, Tỡm iu kin ca m hm s cú cc i, cc tiu VD 12 Cho hm s y 3x x2 1, Kho sỏt v v th hm s 2, Tỡm cỏc im trờn th ca hm s cú ta l nhng s nguyờn VD 13 Cho hm s y x x 1, Kho sỏt hm s 2, Cho d : 2x-y+m = CMR d luụn ct th hm s ti hai im A, B phõn bit vi mi m 3, Tỡm m AB ngn nht VD 14 Cho hm s y 2x cú th (C) 2x 1, Kho sỏt v v th hm s (C) 2, Vit phng trỡnh tip tuyn ca th (C) ti giao im ca (C) vi trc tung VD 15 Cho hm s y 2x x 1, Kho sỏt s bin thiờn v v th ca hm s ó cho 2, Tỡm m ng thng y mx ct th ca hm s ó cho ti hai im phõn bit VD 16 Cho hm s y 2x cú th (C) x 1, Kho sỏt s bin thiờn v v th (C) ca hm s 2, Vit phng trỡnh cỏc ng thng song song vi d: y x v tip xỳc vi th (C) VD 17 Cho hm s y x 3x (C) 1, Kho sỏt v v th (C) ca hm s 2, Da vo th (C) , bin lun theo m s nghim thc ca phng x 3x m 3, Tỡm m phng trỡnh x3 3x m3 3m cú nghim phõn bit 4, Vit phng trỡnh tip tuyn ca (C) ti im cú honh x 5, Vit phng trỡnh ca (C) ti cỏc im cú tung y VD 18 Cho hm s y 4x 3x (C) 1, Kho sỏt v v th (C) ca hm s 2, Da vo th (C) bin lun theo m s nghim thc phng trỡnh : x x m 3, Bin lun s nghim phng trỡnh 4x 3x 4m 3m theo m 4, Vit phng trỡnh tip tuyn ca (C) , bit tip tuyn i qua im M 1, VD 19 Cho hm s y = 2x - 3x - 1, Kho sỏt v v th (C) ca hm s (C) 16 Gia s Thnh c www.daythem.edu.vn 2, Vit phng trỡnh tip tuyn ca (C), bit tip tuyn vuụng gúc vi d1 : y x 2015 3, Vit phng trỡnh ng thng i qua M 2;3 v tip xỳc vi th (C) 4, Tỡm m ng thng d : y mx ct th (C) ti im phõn bit 5, Tỡm m ng thng d : y m x ct th (C) ti im phõn bit VD 20 Cho hm s y = (2 - x )(x + 1)2 (C) 1, Kho sỏt v v th (C) ca hm s 2, Tỡm m th (C) y x m ct th (C) ti im phõn bit 3, Vit phng trỡnh tip tuyn ca (C), bit tip tuyn vuụng gúc vi ng thng d1 : y x 4, Tỡm m ng thng d : y m x ct th (C) ti im phõn bit VD 21 Cho hm s y x3 mx 2(3m 1) x 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 cho x1x2 + 2(x1 + x2 ) = VD 22 Cho hm s y x3 (m 1) x (m 4m 3) x 1, Kho sỏt s bin thiờn v v th ca hm s m = -3 2, Vi giỏ tr no ca m hm s cú C, CT x1; x2 Tỡm GTNN ca x1 x2 2( x1 x2 ) VD 23 Cho hm s y x3 (m 3) x 2( m 1) x 1, Kho sỏt s bin thiờn v v th ca hm s m = 2, Tỡm tt c cỏc giỏ tr ca tham s m hm s cú hai im cc tr vi honh ln hn VD 24 Cho hm s y x3 x mx 1, Kho sỏt hm s vi m = - 2, Tỡm m hm s t cc tr ti x1; x2 cho y ( x1 ) y ( x2 ) x1 x2 VD 25 Cho hm s y x3 (m 1) x 2(m 2) x 1, Kho sỏt hm s vi m = 2, Tỡm m hm s t cc tr ti x1; x2 cho P x1 x2 x1 x2 VD 26 Tỡm m th hm s (Cm): y x3 3mx 3x 3m ct trc Ox ti im phõn bit cú honh l x1, x2, x3 tha x12 x22 x32 15 VD 27 Cho hm s y x3 x (1 m) x m 1, Kho sỏt s bin thiờn v v th ca hm s vi m = 2, Tỡm m th hm s (Cm): y x3 x (1 m) x m ct trc Ox ti im phõn bit cú honh l x1, x2, x3 tha x12 x22 x32 < VD 28 Cho hm s y x x (C) 1, Kho sỏt v v th (C) ca hm s 2, Bin lun theo m s nghim thc ca phng trỡnh x x m 3, Vit phng trỡnh tip tuyn ca th (C) ti im cú honh x 4, Vit phng trỡnh tip tuyn ca th (C) , bit h s gúc ca tip tuyn bng 24, 5, Tỡm m y=m ct (C) ti im phõn bit cú honh lp thnh cp s cng 17 Gia s Thnh c VD 29 Cho hm s y www.daythem.edu.vn 2x (C) x 1, Kho sỏt v v th (C) ca hm s 2, Tỡm m ng thng d : y mx 2m ct (C) ti im phõn bit 3, Vit PTTT ca (C) bit tip tuyn to vi hai ng tim cn mt tam giỏc cú chu vi x x VD 30 Cho hm s y (C) 1, Kho sỏt v v th (C) ca hm s 2, Tỡm m d : y mx 2m VD 31 Cho hm s y ct th (C) ti im phõn bit cú honh dng 3x (C) x 1, Kho sỏt v v th (C) ca hm s 2, Tỡm m d : y mx 2m ct th (C) ti hai im A, B phõn bit Tỡm hp trung im I ca on thng AB 3, Tỡm nhng im trờn th (C) cú to vi honh v tung u l s nguyờn VD 32 Cho hm s y 2x (C) x2 1, Kho sỏt v v th (C) hm s 2, Tỡm phng trỡnh tip tuyn vi (C) ti im M thuc (C) v cú honh x o= 3, Tỡm N cho PTTT ti N ca (C) cỏch tõm i xng I on VD 33 Cho hm s y 2x (C) x3 10 1, Kho sỏt s bin thiờn v v th ( C ) ca hm s 2, Tỡm M cho PTTT ti M ca (C) cỏch tõm i xng I on LN VD 34 Cho hm s y 2x , (C) x 1, Kho sỏt s bin thiờn v v th (C) ca hm s 2, Tỡm m d: y = mx + ct th (C) ti hai im phõn bit M, N: MN VD 35 Cho hm s y 2x C x 11 1, Kho sỏt hm s 2, Gi (d) l ng thng qua A( 1; ) v cú h s gúc k Tỡm k cho (d) ct ( C ) ti hai im M, N v MN 10 VD 36 Cho hm s y 2x (1) x 1, Kho sỏt s bin thiờn v v th ca hm s (1) 2, Tỡm k ng thng d: y kx ct th hm s (1) ti hai im M, N cho tam giỏc OMN vuụng gúc ti O ( O l gc ta ) VD 37 Cho hm s : y 2x x (1) 18 Gia s Thnh c www.daythem.edu.vn 1, Kho sỏt v v th (C) ca hm s (1) 2, Chng minh rng ng thng d: y = 2x + m luụn ct th (C) ti hai im M v N phõn bit vi mi m Xỏc nh m on thng MN ngn nht MT S B I H PHNG TRèNH HAY Những lúc suy t- y (3x 2x 1) 4y 2 y x 4y x 6y 5y 4x 12x 9x (2 y) 2y 2 2x 5x 2y 3y x y y x y 1 y 1 x 2 2 x xy 2y 2x xy y 2(x y) (8 y 6) x (2 y 2)(y x 3) x y x 2y 2( x 4y) y xy 2y 34 15x 2 2x 5xy 3y 49(2y 3x 1) 2 x 2xy 10y 637 2y3 12y2 25y 18 (2x 9) x 2 3x 14x 3x 4y y x y3 x y x y x x 9(y 1) 2x x x y y x x3 x 11 x y x y(x 1) 2x 5xy y2 2 y( xy 2y 4y xy) 2 2 x(4y 3y 5y x ) y (x 4y 8) 10 x 12 2x 2y y 2x 4x (y y ) 12 x x y xy 2xy x y MT S PHNG TRèNH HAY 1, (3x + 1) 9x + 6x + - x + = 4x 16x + 2, - 10x + 12x - 5x + = 2x 7x - 7x + 2x 3, - 2x + 10x - 17x + = 2x 5x - x 4, x + 3x + 4x + = (3x + 2) 3x + 5, 8x - 36x + 53x - 25 = 3x - 6, 7, 27x -27x + 13x - = 2x-1 9, (x - 1) 11, x ( x+ 1- 2x + - = x+ 8, 4x + 18x + 27x + 14 = 4x+ - x - - (x - 5) x - - 3x + 31 = 10, (s inx - 2)(sin x - s inx + 1) = 3 s inx - + x+ 1+ ) x+ = ( + + x2 ) ( 12, 2x + ) 4x + = x + x2 + 13, 4x + x + (x - 3) - 2x = 14, x3 x2 3x 2(3x 1) 3x 15, x x x x x 16, x - 15x + 78x - 141 = 2x - ( ) 19 Gia s Thnh c www.daythem.edu.vn 18, (x + 6) + x + - x - 12x - 48x - 64x - x - 4x = 17, x + 6x - 171x - 40 (x + 1) 5x - + 20 = 19, 27x + 54x + 39x + 10 = (x + x + 2) x + x + 20, 2x + (x + 1) x + 2x + + (x + 2) x + 4x + + = ( ) 21, x + 4x + = x + 23, (x + 5) x + + = x2 + 22, (x - 1) - 8x + x + 32x + 24, x - 3x - 2x - - 3x + ( ) x - - 23x = - 23 3x + 3x + = 26, x + 2x - = (x + 1) x - 2x+ ổ1 x ữ 4(1 + + 4x ) ỗ ữ = 27, ỗỗ ữ ỗỗố x x + 1ữ ữ x + + x + 3x + ứ 28, x+ = (x + 2) x + x3 + x 29, x (4x + 1) - (x + x + 1) 2x + 2x + 1= 30, x + 9x - 156x-144 = 20 (x + 2) 5x + 25, 17 - x + 6x - 2x = 14 - 2x + 3x - x ( x+ 2- ) 20 [...]... 2 1 2 Cho ba s thc a, b, c tha món a b c 1 CMR: a 2 1 b2 1 c 2 1 10 a b c a c b c b a l cỏc s thc dng CMR: 2 2 2 2 c b a b2 a c a 2 b c 2 VD 10 Cho a,b,c Bài: khảo sát hàm số 1, Kho sỏt v v th hm s *) TX: D *) SBT: CBT: Tớnh y Du y v suy ra CBT 14 2 2 3 5 Gia s Thnh c www.daythem.edu.vn Cc tr Gii hn Bng bin thiờn, *) V th 2, Bin lun s nghim phng trỡnh da vo th... GTNN P y 1 x 1 x y 3 x y xy 3 x y VD 13 Cho cỏc s thc dng tho: x + y = 1 Tỡm GTNN ca: P 1 x 1 y VD 15 Cho y 0, x 2 x y 12 Tỡm GTLN, NN ca: P=xy + x + 2y +17 Cho x, y l cỏc s thc khụng õm thay i v tha món iu kin 4( x y xy ) 1 2( x y ) Tỡm giỏ tr ln nht ca biu thc P xy x y x 2 y 2 VD 22 Cho cỏc s thc khụng õm x, y, z tho món x 2 y 2 z 2 3 VD 21 2 2 Tỡm giỏ tr ln nht ca biu... Thnh c www.daythem.edu.vn 1, Kho sỏt v v th (C) ca hm s (1) 2, Chng minh rng ng thng d: y = 2x + m luụn ct th (C) ti hai im M v N phõn bit vi mi m Xỏc nh m on thng MN ngn nht MT S B I H PHNG TRèNH HAY Những lúc suy t- 3 2 y (3x 2x 1) 4y 8 1 2 3 2 2 y x 4y x 6y 5y 4 3 2 4x 12x 9x 1 (2 y) 2y 1 2 2 2 2x 5x 3 2y 3y 1 0 9 x 1 2 y 1 y 4 3 x 1 y 1 1 1 y 1 1 x 1... 2 5xy y2 1 8 2 2 y( xy 2y 4y xy) 1 3 2 2 2 2 2 x(4y 3y 5y x ) y (x 4y 8) 10 2 x 12 2x 2y 2 y 4 2x 4x 2 1 (y 1 y 2 ) 1 12 x x y xy 1 2xy x y 1 MT S PHNG TRèNH HAY 1, (3x + 1) 9x 2 + 6x + 2 - x + 1 = 4x 16x 2 + 1 2, - 10x 3 + 12x 2 - 5x + 1 = 2x 2 3 7x 3 - 7x 2 + 2x 3, - 2x 3 + 10x 2 - 17x + 8 = 2x 2 3 5x - x 3 4, x 3 + 3x 2 + 4x + 2 = (3x + 2) 3x + 1 5, 8x 3