Khi m = 0 thi dd thi cd TCD va TCN vudng gdc (loai). Khi m * 0 thi do thi cd tiem can dung: x = -3m va tiem can xien: y = mx - 2. Hai tiem can hop nhau goc 45° khi tiem can xien hop vdi true hoanh mot gdc 45° <=> m = ± 1. Vi du 7: Tim tat ca cac diem M thuoc (C): y x + 1 x-l sao cho khoang each tir M den giao diem hai dudng tiem can cua (C) ngan nhat. Giai x + 1 x-l Do thi (C): y can la 1(1; 1). x + 1 Ta cd M(x; ) e (C) nen khoang each: x - l cd TCD: x = 1, TCN: y = 1 nen giao diem 2 tiem I M = 1(x-l) 2 + | 1 Dau = khi (x-l ) x - l 2_ 4 (x-l) 2 + - >4 (x-l) 2 <=>(x- l)2 = 2c=>x= 1 ± yfe . (x-l ) Vay cd hai diem M thoa man bai toan: Mi(l + 42 ; 1 + V2), M2(l- yfe ; 1-V2) Vi du 8: Cho ham so y = mx + — Tim m de ham so cd cue tri va khoang x each tir diem cue tieu den tiem can xien bang Giai 2 1 D = R \ {0} , y' = m x " Ta cd : y' = 0 o mx2 = 1 x Xet m = 0: 0 = 1 (loai). Xet m ^ 0: x2 = —, dieu kien cd cue tri la m > 0, m ta cd y' = 0 <=> x = ± BBT: 1 X -CO 0 Vm +00 y' + 0 - - 0 + y Diem cue tieu A 1 ;2Vm Do thi cd tiem can xien d: y = mx <=> mx - y = 0. -BDHSG DSGT12/1- 99 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Vm' + l V2 o 2 m = m 2 +lo(m-l) 2 = 0om= l (chon) x + (1 — m)x — 2 V j du 9: Cho ham so y = (C m ). Tim m de tiem can xien ciia x + 1 (C m ) tao vdi cac true toa do thanh mot tam giac cd dien tich bang 18. Giai Hamsdy = x-m + —-,D = R\ {-1}. x + 1 Ta cd lim(y-(x-m)) = 0 nen tiem can xien d cua (C m ) cd phuong trinh x->±x y = x - m. Giao diem cua d vdi Ox: A(m; 0), giao diem cua d vdi Oy: B(0; -m) Dien tich tam giac OAB la S = — m 2 Dieu kien S = 18 <=> -m 2 = 18 <=> m = ±6. 2 V i du 10: Chung minh rang tren do thi (C): y = 2x + 1 - — khdng co diem x nao tai do tiep tuyen song song vdi tiem can xien cua do thi ham so da cho. Giai Dd thi (C) cd TCX: y = 2x + 1, he sd gdc a = 2. Ta cd he so gdc cua tiep tuyen la dao ham tai dd nen k = f '(x) = 2+ \ >2 , Vx ^ O suy ra dpcm. x C. BA I LUYE N TAP Bai 1: Tim tiem can dung cua do thi: . x + 1 , 4x + 3 7r. a) y = b) y = —r — c) y = tan(x - - ) x + 3x x+ 1 2 DS: a) x = 0 va x = - 3 b) khong cd Bai 2: Tim tiem can dung cua do thi: x , . x - 2 . x2 - m a )y=—— 2 7 h ) y = ^- , c ) y = — T T x+m+ m + 1 x+4x- m x- 2 DS: a) khong cd c) m = 4: khdng cd; m * 4: x = 2 Bai 3: Tim tiem can ngang ciia do thi : x - l , \ x 2 - l a)y=—2—^ 7 b ^ y = 7 o c) y = v'4-x - x2 + 3x + 1 5x + 3 100 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
DS: a) y = 0 b) khong cb Bai 4: Tim tiem can ngang ciia db thi: , x3 Vx2 -2 x a) y = ——- b) y = —•—-— c) y = arctan x x - l 1 + 3x DS: c) x = ± - 2 Bai 5: Tim tiem can xien cua do thi: ux .. x 2 - 1 x x3 -4 x + sinx x2 a) y = x - 2 + b) y = c) y x + 1 x DS: b) y = x + 2 c) y = x Bai 6: Tim tiem can xien cua dd thi: a)y = 3x + Vx2 + 6x + 1 b) y = V3x2 -x3 c) y = x + sin x BS: a) y = 2x - 3 va y = 4x +3 c) khdng cd Bai 7: Tim cac tiem can cua dd thi: , 6x + 5 3x2 + 4x + 7 a) y = b) y = x - 9 , V x- 2 v (x2 - l)cosa + xcos a C ) y = '- x - cosa Bai 8: Tim cac tiem can cua dd thi: a)y = x./^-!— b)y = x- l + Vx2 -2 J c)y = x + 2y / 9 - x 2 DS: a) x = 1 va y = x +1 b) y = 0 va y = 2x Bai 9: Tim cac tiem can cua do thi : , mx2 -2(m + 3)x , x mx3 -l a) y = b) y = — c) y = — x - l x 2 - m x 2 - 3 x + 2 Bai 10: Tim m de tiem can xien cua do thi : s 2x2 +(m + l)x-3 TT /1 i \ a) y = qua H (1,1) x + m , x x2 + mx -1 ,. „ ^ , o) y = tao voi 2 true toa do thanh tam giac cd S = 1. x - l DS: a) m = 2 b) m = - 1 ± ^2 Bai 11: Tim diem M tren dd thi y = — - 2 + each deu 2 tiem can 3 x- 2 -BDHSG DSGT12/1- 101 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Bai 12: Tim diem M tren dd thi (H) : y = 3 cd tdng khoang each den 2 x - 3 tiem can be nhat. DS: M(0;l)hoacM(6;7) Bai 13: Tim a de khoang each tu goc toa do den tiem can xien cua do thi x2 cosa + 2xsina - 1 .. ,, . x (C) : y = la Ion nhat. x - 2 Bai 14: Chung minh tich cac khoang each tu mot diem tuy y tren do thi den hai tiem can la so khong doi. 3x2 +5x- l x2 + 3x - 2 2x- 3 Bai 15: Cho dd thi (C) : y = x - l a) Tinh gdc giua 2 tiem can. b) Chung minh do thi khong cd tiep tuyen nao song song tiem can. DS: a) 45° Bai 16: Tim tam doi xung cua do thi ham so 3x- 4 , . x 2 +3mx + 3m2 a) y = b) y = ,m* 0 x + 5 x + 2m H D: Tam doi xung la giao diem 2 tiem can. Bai 17: Tim m de do thi y = 2x + ^m ———2m + - nhan 1(2; 1) lam tam x - 2 doi xung. DS: m = -4. 102 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
§ 5 . KHA O SA T V A V E HA M D A THU C A. KIEN THlfC CO BAN Sff do chung ve khao sat va ve do thj ham da thuc: Gom 3 budc: Buocl: Tap xac djnh - Tap xac dinh D = R - Xet tinh chan, le neu cd. Budc 2: Chieu bien thien - Tinh cac gidi han. - Tinh dao ham cap mot, xet dau - Lap bang bien thien rdi chi ra khoang ddng bien, nghich bien va cue dai, cue tieu. Budc 3: Ve do thi - Tinh dao ham cap hai, xet dau de chi ra diem udn cua ham da thuc. - Cho vai gia tri dac biet, giao diem vdi hai true toa do. - Ve dung do thi, luu y ham bac 3 cd tam doi xung la diem udn, ham trung phuong la ham sd chan nen do thi nhan true tung la true ddi xung. Diem uon cua do thi : Diem U(xo; f(xo)) dugc goi la diem uon cua dudng cong (C): y = f(x) neu ton tai mot khoang (a; b) chua diem xo sao cho mot trong 2 khoang (a, xo), (xo, b) thi tiep tuyen tai diem U nam phia tren do thi con d khoang kia thi tiep tuyen nam phia dudi do thi. Phuong phap tim diem udn: Cho y = f(x) cd dao ham cap 2 mot khoang (a; b) chua diem xo. Neu f "(xo) = 0 va f "(x) ddi dau khi x qua diem xo thi U(xo; f(xo)) la diem udn cua dudng cong (C): y = f(x). Chii y: Neu y =p(x).y'' + r(x) thi tung do diem udn tai xo la yo = r(xo) - Tinh ldi, 1dm cua duong cong ( md rdng): Cho ham sd y = f(x) cd dao ham cap 2 tren (a,b) - Neu f "(x) > 0, Vx e (a,b) thi dd thi 16m tren (a,b) - Neu f "(x) < 0, Vx G (a,b) thi dd thi ldi tien (a,b) - Diem udn la diem ngan each phan loi va phan 1dm. - Neu f loi tren doan [a,b] thi GTLN = max{f(a); f(b)} - Neu f 1dm tren doan [a,b] thi GTNN = min{f(a); f(b)} -BDHSG DSGT12/1- 103 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
B. PHAN DANG TOA N DANG 1: HAM BAC BA Dang do thi ham bac 3: y = ax3 + bx2 + cx + d, a * 0 Tam ddi xiing la diem udn. V i du 1: Cho ham sd y = 2x3 - 6x + 1 a) Khao sat su bien thien va ve do thi (C) cua ham sd da cho. b) Dua vao do thi (C), bien luan theo tham so m sd nghiem phan biet cua phuong trinh 2x3 - 6x + 1 - m = 0. Giai • Tap xac dinh D = R. . Sir bien thien: lim y = -co, lim y = +oo a) 2 Dao ham: y' = 6xz - 6, y' = 0 « - x = - 1 hoac x = 1. y'>0ox e (-oo; -1) u (1; +oo); y' < 0 <=> x e (-1 ; 1) Bang bien thien: -1 1 +00 0 0 +C0 4 ) va (1; +QO), nghich bien tren - 1 , yco = 5 va dat cue tieu tai Ham sd ddng bien tren moi khoang (-co; khoang (-1 ; 1). Ham sd dat cue dai tai x : x = l,ycT = -3 . Dd thi: y" = 12x, y" = 0 <=> x = 0 nen diem udn 1(1; 0) la tam ddi xiing. Dd thi cat true Oy tai diem (0;. 1). b) Phucmg trinh da cho tuong duong 2x3 - 6x + 1 = m. Do dd, so nghiem cua phuong trinh da cho bang sd diem chung ciia do thi (C) va dudng thang y = m. Dua vao do thi (C), ta duoc: - Neu m > 5 hoac m < -3 thi phuong trinh cd 1 nghiem. - Neu m = 5 hoac m = -3 thi phuong trinh cd 2 nghiem. - Ngu -3 < m < 5 thi phuong trinh cd 3 nghiem phan biet. 104 -BDHSG DSGT12/1- y ' 5 / i / y = m i 1 / \ 1 / • 1 - i o / -3 \ \ 1 x Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Vi du 2: Cho ham so y = x3 + 3mx2 + (m + l)x + 1 (1) a) Khao sat sir bien thien va ve do thi (C) khi m = -1 . Chung minh (C) co tam doi xung b) Tim cac gia tri cua m de tiep tuyen cua dd thi ham sd tai diem cd hoanh do x = - 1 di qua diem A(l ; 2). Giai a) Khi m = - 1 thi y = x3 - 3x2 + 1. . Tap xac dinh D = R. . Su bien thien lim y = -oo va lim y = +oo X—cu X—>+00 y' = 3x2 - 6x, y' = 0 co x = 0 hoac x = 2. Bang bien thien Ham sd dong bien tren cac khoang (-oo; 0) va (2; +oo), ham sd nghich bien tren khoang (0; 2). Ham so dat CB(0; 1), CT(2; -3). . Dd thi: y" = 6x - 6, y" = 0 co x = 1 nen do thi cd diem uon 1(1; -1). Chox = 0=>y = 1. Chuyen he true bang phep tinh tien fx = X + 1 OI: [y = Y- l The vao (C) thanh: Y - 1 = (X + l) 3 - 3(X + l) 2 + 1 o Y = X3 - 6X Ta cd Y = F(X) = X 3 - 6X la ham sd le =0 dpcm. b) Goi M la diem thuoc do thi ham sd (1) cd hoanh do x = -1 , suy ra M(-l ; 2m - 1). Ta cd y' = 3x2 + 6mx + (m + 1) ; y'(-l) = 4 - 5m. Tiep tuyen d cua dd thi ham sd da cho tai M(-l ; 2m - 1) cd phuong trinh la: y = (4 _ 5m)(x+ 1) + 2m - 1. 5 Tiep tuyen d di qua A(l ; 2) khi va chi khi 2 = (4 - 5m)2 + 2m-lom= - -BDHSG DSGT12/1- 105 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
V i du 3: Khao sat su bien thien va ve do thi cua ham sd a) y = -x 3 + 3x2 - 4x + 2 b) y = x3 - 3x2 + 3x + 1. Giai a) • Tap xac dinh D = R . Su bien thien lim y = -co va lim y = +oo X-*—OC X—>+x Ta cd y' = -3x2 + 6x - 4 < 0, Vx nen ham sd nghich bien tren R. Ham so khong cd cue tri. Bang bien thien X —CO +00 y' - y +00^^ ^ 00 21 \ l 0 -2 l \ .Do thi: y" = - 6 x + 6, y" = 0 co x = 1 nen do thi cd diem uon 1(1; 0). Cho x = 0 => y = 2. Cho y = 0 co -x 3 + 3x2 - 4x + 2 = 0 c> (x - l)(x2 — 2x + 2) = 0 <=> x = 1. b) • Tap xac dinh D = R. • Su bien thien: lim y = -co va lim y = +oo X—MKJ X->+cr 2 2 ' ' ' Ta cd y' = 3x - 6x + 3 = 3(x - 1) > 0, Vx nen ham sd ddng bien tren R, ham sd khdng cd cue tri. Bang bien thien: X —00 \ +00 y' + 0 y +00 -00--— .Do thi: y" = 6x - 6, y" = 0 co x = 1 nen do thi cd diem udn 1(1; 2). Chox = 0=> y = -1 . V i du 4: Cho ham so y = -x 3 + 3x2 + mx - 2 (1), m la tham sd. a) Khao sat su bien thien va ve do thi ciia ham so (1) khi m = 0. b) Tim cac gia tri m de ham so (1) nghich bien tren khoang (0; 2). Giai a) Khi m = 0 ham so trd thanh y = -x 3 + 3x2 - 2. Bai 648 (ve) b) Tacdy' = -3x2 + 6x + m Ham so nghich bien tren (0; 2) khi va chi khi y' < 0, Vx e (0; 2). co m < 3x2 - 6x, Vx e (0; 2) 106 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Xet ham so g(x) = 3x2 - 6x voi x e (0; 2). Ta co g'(x) = 6x - 6, g'(x) = 0 o x = 1. X 0 1 2 g'(x) + 0 g(x) Tii bang bien thien suy ra cac gia tri can tim la m < -3. Vi du5: Cho ham so y = - x3 + (m - l)x2 + (m + 3)x - 4 3 a) Khao sat su bien thien va ve do thi (C) cua ham sd khi m = 0. b) Xac dinh m de ham sd dong bien tren khoang (0; 3). Giai a) Khim = 0thiy = -ix3 -x2 + 3x-4. ; 3 . Tap xac dinh D = R . Su bien thien: lim y = +oo va lim y = -oo y'= -x2 - 2x + 3; y' = 0 <=> x = 1 hoac x = -3. y' > 0 o x £ (-3; 1): ham so ddng bien tren (-3; 1) y' < 0 o x e (-co; -3) u (1; +oo): ham sd nghich bien tren mdi khoang (-oo; -3) va (1; +oo). Bang bien thien: —00 -3 1 +00 +O0 -111 -13' Ham sd dat cue dai tai: x = 1, yco = y(l) - - Ham sd dat cue tieu tai: x = 3, ycr = y(-3) = -13. . D6 thi: y" = -2x - 2, y" = 0 * o x = - 1 nen dd thi cd diem 23 udn I(-l ; ) la tam doi xung. 3 , Dd thi ham sd cat true Oy tai diem (0; -4). b)~ y' = -x 2 + 2(m - l)x + (m + 3); A' = m2 - m + 4 > 0, Vm nen y' ludn hai nghiem phan biet. -snmr; nscrnil- 107 CO Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Dieu kien dong bien tren (0; 3): y' > 0, Vx e (0; 3) m >-3 y'(0) = m + 3> 0 y'(3) = 7m-12> 0 1 12 <=> < 12 12 <=>m> — m > — 7 7 Vay vdi m > — ham sd da cho dong bien tren khoang (0; 3). Vi du 6: Cho cac do thi (Cm): y = — x3 - mx2 - 3mx - — 3 3 a) Tim cac diem cd dinh. b) Khao sat va ve do thi (C) khi m = 1. Suy ra dd thi (C): y 1 7 9 n 5 - x 3 - x -3x- - 3 3 Giai a) Goi M(XQ; y0 ) la diem cd dinh cua cac do thi (C m ): y0 = — x3 - mx2 - 3mxo - - . Vm 3 3 ° 3 1 5 <=>y0=-m(x2 +3x o ) + -x 3 -- , Vm <=> { xl + 3x = 0 5 » 3"° 3 1 3 y„ =-x_ - x Q = -3, y0 = -32 Vay cac do thi di qua 2 diem cd dinh: M\ 0;— | va M2 v 3 b) Khi m = 1 thi y = -x2 - x2 - 3x -- 3 3 - Tap xac dinh D = R • Su bien thien lim y = -00 va lim y = +co X—V—DO X-»+ae y' = x2 - 2x - 3, y' = 0 <=> x = -1 hoac x = 3. Bang bien thien -3 -32 ^ -co - 1 +00 +00 -32/3 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Ham sd dong bien tren cac khoang (-co; -1) va (3; +00); nghich bien tren khoang (-1 ; 3). -32 Ham sd dat cue dai tai x = -1 ; yco = 0 va dat cue tieu tai x = 3; yet = —— Dd thi: y''-2x-2 , y" = 0o x = l nen do thi cd diem uon I l;-i 2 3 , Ta cd y x 3 - x 2 -3x - - x 3 - x 2 -3x - khi x >5 x 3 - x 2 -3x-- | khix < nen do thi (C) giu nguyen phan dd thi (C) khi x > 5 va lay ddi xung phan x < 5 cua (C) qua Ox. Vi du 7: Cho ham so y = x3 - 3x2 - 9x. a) Khao sat va ve do thi ham sd. Tinh khoang each giua cue dai va cue tieu. b) Bien luan theo m sd nghiem cua phuong trinh: x3 - 3 3x2 - 9x = m3 - 3m2 - 9m. Giai a) . Tap xac dinh D - R . Su bien thien y' = 3x2 - 6x - 9, y' = 0 co x = - 1 hoac x = 3. -BDHSG DSGT12/1- 109 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
BBT X —00 - 1 3 +00 y' + 0 0 + y —CO 5 -27 +00 Do thi co cue dai A(-l ; 5), cue tieu B(3; -27) nen khoang each AB =Vl 6 + 1024 =W6 5 • Do thi: y1 ' = 6x - 6, y'' = 0 <=> x = 1 nen do thi co diem udn I(l;-ll) . Cho x = 0 thi y = 0. y i b) Dat f(x) = x3 - 3x2 - 9x thi phuong trinh: f(x) = f(m). Ta cd y = 5 co x = - 1 hoac x = 5; y = -27 khi x = -3 hoac x = 3. Dua vao do thi, ta cd: Khi m < -3 hoac m > 5 thi PT cd 1 nghiem. Khi m = -3 hoac m = - 1 hoac m = 5 hoac m = 3 thi PT cd 2 nghiem. Khi -3 < m < 5, m * -1 , m ^ 3 thi PT cd 3 nghiem. V i du 8: Cho ham so y = — x3 - 3x. 4 a) Khao sat su bien thien va ve do thi (C) cua ham sd da cho. b) Lap phuong trinh cac tiep tuyen song song vdi d: y = 6x. Suy ra so nghiem phuong trinh: — x3 = 9x - 12 V3 + m theo m. 4 Giai a) • Tap xac dinh D = R. Ham so le. ' 3 . Su bien thien y1 = — x 4 3,y' = 0co x = ±2. 110 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
BBT X —00 -2 2 +00 y' 0 0 + y —00 4 - 4 +O0 Ham so dong bien tren (-oo; -2), (2; +co), nghich bien tren (-2; 2). Ham so dat cue dai tai (-2; 4), cue tieu tai (2; -4). .D6thi:y " = -x , y" = Ocox = 0 2 y | nen do thi nhan gdc O lam diem uon. Cho y = 0 co x = 0 hoac x = ±27 3 b) Tacdf'(x) = 3 x 2 - 3 = 6 4 => x2 = 12 xD = ±2 V3 Khi x0 = -2 73 thi tiep tuyen y = 6x+ 12 73 Khi x0 = 2%/3 thi tiep tuyen y = 6x - 12 73 PT: -x 3 = 9x-12x/3 + m o-x 3 - 3x = 6x - 12 73 + m 4 4 Cac dudng thang y = 6x - 1273 + m song song hoac trung vdi cac tiep tuyen y = 6x + 12 73 Dua vao do thi ta cd: Neu m < 0 hoac m > 24 73 thi PT cd 1 nghiem. Neu 0 < m < 24 73 thi PT co 3 nghiem. Neu m = 0 hoac m = 24 73 thi PT co 2 nghiem. Vj du 9: Cho ham so y = x3 - 3x2 + mx (1) a) Khao sat va ve dd thi ham so khi m = 0. b) Tim tat ca cac gia tri cua tham so m de ham sd (1) cd cue dai, cue tieu va cac diem eye dai, cue tieu cua do thi ham sd doi xung nhau qua dudng thang (d): x - 2y - 5 = 0. Giai a) Vdi m = 0 thi y = x3 - 3x2 • Tap xac dinh: R . Su bien thien: limy = -00; limy = +00 X —> — X X—>+oq y' = 3x2 - 6x, y' = 0 ci> x = 0 hoac x = 2. -BDHSG DSGT12/1- 1 1 ) Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Bang bien thien: —CO +00 + +00 —CO Ham so dong bien tren (- co;0), (2, + co) , nghich bien tren (0,2) va co diem CD(0; 0), CT(2; -4) .D6 thi y" = 6x-6 , y" = 0 » x = 1. Diem uon 1(1; -2) b) y = x3 - 3x2 + mx ; y' = 3x2 - 6x + m Dieu kien de ham so cd cue dai, cue tieu Mjfe ; yx), M2 (x2 ; y2 ) la y' = 0 cd hai nghiem phan biet: A' = 9 - 3m > 0 c=> m < 3. Ta cd: f(x) = (-x - - ) f' (x) + (-m-2) x + - m 3 3 3 3 x i, X2 la hai nghiem cua y' = 0 xl + x2 =2 m 2 1 2 1 Yi = f(xi) = ( - m - 2)xj + - m, y2 = f(xz) = ( - m - 2)x2 + - m Duong thing (d): x - 2y - 5 = 0 co VTCP v* = (2; 1) Goi I(x0 ; y0 ) la trung diem cua M 1 ; M 2 x o = T;(X I + X 2 ) = 1 y0 =-(y i + y 2 ) = m- 2 1(1; m - 2) Do Mi , M 2 doi xung nhau qua (d) nen: Ie(d ) l-2(m-2)- 5 = 0 iM^. v =0 [2(x 2 - x 1 )fy 2 - y 1 = 0 Tu dd giai duoc m = 0. Vi du 10: Cho (C): y = x3 - 4x2 a) Khao sat va ve do thi (C). b) Chung minh (C) tiep xuc (P): y = x2 - 8x + 4. 112 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Giai a) • Tap xac djnh D = R. • Su bien thien lim y = -co va lim y = +co y' = 3x - 8x, y' = 0 co x = 0 hoac x = - 3 BBT X —CO 0 8/3 +00 y' + 0 0 + y —OO 0 \ * 256 / 27 +00 . Do thi: y" = 6x - 8, y" = 0 4 <=> x = — nen do thi cd 3 w x T(4 128^ diem uon 1 —; . 1.3 27 ) y = 0 co x = 0 hoac x = 4. b) Hai do thi (C) va (P) tiep xuc khi he sau cd nghiem: jf(x) = g(x) jf'(x) = g'(x) ' fx3 -5x2 +8x-4 = 0 CO { [3x2 -10x + 8 = 0 x = 1 hay x = 2 4 <o x = 2 x = — hay x = 2 3 " Vay 2 dd thi tiep xuc nhau tai diem A(2; -8). x 3 -4x 2 3x2 -8 x = 2x- 8 DANG 2: HAM TRUNG PHUONG Dang do thi ham trung phuong: y = ax4 + bx2 + c, a * 0 -BDHSG DSGT12/1- 113 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
2. CAC BA I TOA N DIEN HIN H V i du 1: Cho ham so y = x4 - 8x2 + 7. a) Khao sat sir bien thien va ve dd thi cua ham so. b) Tim cac gia tri cua tham sd m de dudng thang y = mx - 9 tiep xuc vdi do thi cua ham sd. Giai a) • Tap xac dinh D = R. Ham so chan. - Su bien thien: lim y = +co. x->±vy' = 4x3 - 16x = 4x(x2 -4 ),y ' = 0o x = 0 hoac x = ±2. Bang bien thien: X —00 -2 0 2 +00 y' - 0 + 0 - 0 + y +00 -9"" " 7 *-9^ +O0 Ham sd ddng bien tren cac khoang (-2; 0) va (2; +co), nghich bien tren cac khoang (-co; -2) va (0; 2). Ham so dat CD(0; 7), dat CT(-2; -9), (2; -9). • Dd thi: y" = 12x 2 - 16, y" = 0 2 . 4 i , . o x = ±—j= nen do thi co V3 hai diem uon -n) o S ' 9 Cho x = 0 => y = 7, cho y => x = ±1 hoac x = + V7 b) Duong thang y = mx - 9 tiep xuc vdi (C) khi va chi khi he phuong trinh sau cd nghiem: J x 4 -8x 2 + 7 = mx- 9 (1) [4x 3 -16 x = m (2) Thay (2) vao (1) ta duoc: x4 - 8x2 + 7 = (4x3 - 16x)x - 9 c=> 3x4 - 8x2 - 16 Thay x = +2 vao (2) thi m = 0 la gia tri can tim. V i du 2: Cho ham so y = 2x4 - 4x2 a) Khao sat su bien thien va ve dd thi (C) cua ham so. b) Vdi cac gia tri nao cua m, phuong trinh x2 1 x2 - 2 nghiem thuc phan biet? 0 o x = ±2. m cd dung 6 114 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Giai a) - Tap xac dinh D = R. Ham so chan. . Sir bien thien: y' = 8x3 - 8x; y' = 0 <=> x = 0 hoac x = ±1. Ham sd nghich bien tren (-oo, -1) va (0; 1), dong bien tren (-1 ; 0) va (1; +oo). Ham sd dat cue tieu tai x = ±1, ycT = -2 ; dat cue dai tai x = 0, yco = 0. lim y = lim y = +co X-V-OO X—>+oo Bang bien thien X —00 - 1 0 1 +00 y' - 0 + 0 - 0 + y +00 -1' ^ 0 A - 2 X +00 . Do thi: y" = 24x2 - 8, y" = 0 <=> x = ±- = nen do thi cd hai diem udn b) Ta cd x2 1 x2 - 2 | = m <=> I 2x4 - 4x2 1 = 2m. Phuong trinh cd dung 6 nghiem thuc phan biet khi va chi khi dudng thang y = 2m cat do thi (C) ciia ham so y = I 2x4 - 4x2 1 tai 6 diem phan biet. -BDHSG DSGT12/1- 115 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
T - lo 4 A 2| f 2x4 -4x 2 khi|x|>V 2 . . Ta co y = 12x - 4x | =< 1 1 nen do thi (C) [-(2x4 -4x 2 ) khi |x|< V2 dugc suy tu do thi (C) bang each giu nguyen phan do thi d phia tren Ox, con phan phia dudi Ox cua (C) thi lay doi xung qua Ox. Dua vao do thi, yeu cau bai toan duoc thoa man khi va chi khi: 0<2m<2<=>0<m<l . V i du 3: Khao sat su bien thien va ve do thi cua ham sd ' " 4 .2 3 a) y = -x 4 - 2x2 + 5 b) y Giai a) . Tap xac dinh D = R. Ham so chan . Su bien thien lim y = -co va lim y = + x -CO y' = -4x3 - 4x = -4x(x2 + 1), y' = 0 co x = 0 BBT X —oo 0 +G 0 y' + 0 y 5 -00^^ J -CO Ham sd ddng bien tren khoang (-co; 0) va nghich bien tren khoang (0; +co). Ham so dat cue dai tai diem x = 0: yco = 5. • Do thi: y" = -12x2 - 4 < 0, Vx nen do thi khong cd diem uon. Choy = 0=o x = +v / V6- l b) • Tap xac dinh D = R. Ham so chan. • Su bien thien: lim y = +co. y' = 2x3 + 2x = 2x(x2 + 1), y' = 0 co x = 0. BBT X —CO 0 +00 y' - 0 + y +00 +00 ^-3/2 " Ham so dong bien tren khoang (0; +co), nghich bien tren khoang (-co; 0) va dat 3 cue tieu tai (0; —) . 2 . Dd thi: y" = 6x2 + 2 > 0, Vx nen do thi khong cd diem uon. 116 -BDHSG nsr.TU/lDownload Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Giao diem vdi true tung (0; -—) , giao diem vdi true hoanh (-1 ; 0) va (1; 0). 1 3 Vi du 4: Cho ham sd y = — x4 + — mx2 4 2 (D a) Khao sat va ve do thi cua ham sd khi m = 1. b) Tim m de do thi ham so (1) cd 3 cue tri la 3 dinh cua tam giac deu. Giai a) Khim = 1 thi y = --x4 + -x 2 ' 4 2 . Tap xac dinh D = R. Ham sd chan. • Su bien thien: lim y = -co. X—»+x y' = -x3 + 3x = x(3 - x2 ) = 0 o ± S hoac x = 0. y'>0cox = -\/3 hoac 0 < x < ~J3 Ham sd ddng bien trong cac khoang (-co; - %/3 ) va (0; ^3 ). y' < 0 co ->/3 <x<0 hoac V3 < x. Ham so nghich bien trong cac khoang (- V3 ; 0) va (V3 ; +co). BBT: X -co -V3 0 V3 +oo y' + 0 - 0 + 0 - y 9/4 9/4 -:r/ 0 ^-o o Do thi: y" = -3x2 + 3, y" = 0c=> x = ±1 nen do thi cd 2 diem udn ±1 Dd thi cat true tung tai diem (0; 0), cat true hoanh tai ba diem (± V6 ; 0), ~k (0; 0). b) y' = -x3 + 3mx = -x(x2 - 3m) y' = 0 <=> x = 0 hoac x2 - 3m. Dieu kien do thi (1) cd 3 cue tri la 3m > 0 c=> m > 0 ( , 9 \ f Khi do 3 diem cue tri: O(0; 0), A -Sm;-m2 , B I 4 ) fOA = OB Tam giac OAB deu co ^ Ar i oO A = AB OA = AB 9 2 3m; —m 4 1 co m3 = J3m+ — m =2V3m <=> 3m + — m = 12m 16 16 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
<=> m3 = — o m = — V6 (chon). 9 3 x4 V j du 5: Cho ham sd y = a + bx2 (a va b la tham so). 4 a) Khao sat va ve do thi (C) cua ham sd khi a = 1, b = 2. Suy ra sd nghiem cua phuong trinh: 1 + 2x2 - m theo tham so m. b) Tim a va b de ham so da cho dat cue dai bang 4 khi x = 2. Giai x 4 a) Khi a = 1 va b = 2 ta cd ham so y = 1 + 2x2 4 • Tap xac dinh D = R. Ham sd chan. • Su bien thien: lim y = -co y = 4x - x3 = x(4 - x2 ), y' = 0 o x = 0 hoac x = ±2. Bang bien thien X —00 - 2 0 2 +00 y' f 0 - 0 + 0 - y 5 . ^ 1 / r 5 + 00 -OC Ham sd ddng bien tren (-co; -2) va (2; +co) va nghich bien tren (-2; 0) va (2; +oo). Ham sd dat CD tai (+2; 5) va CT tai (0; 1). . D6 thi y" = 4 - 3x2 , y" = 0 2 <=> x = +—= nen do thi cd 2 2 291 diem uon &' 9 y So nghiem cua phuong trinh: 1 + 2x2 - — m bang so giao diem cua dudng thang y = m va dudng cong (C). Dua vao do thi tren ta cd: Neu m = 5 hoac m < 1 thi phuong trinh cd 2 nghiem. Neu m = 1 thi phuong trinh cd 3 nghiem. Neu 1 < m < 5 thi phuong trinh cd 4 nghiem. Neu m > 5 thi phuong trinh vo nghiem. b) Ham so y = f(x) = a + bx2 - — dat cue dai tai diem (2; 4) 118 -BDHSG DSGTW1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
f(2 ) = 4 f'(2) = 0 co f "(2)<0 a + 4b - 4 = 4 4b- 8 = 0 -12 + 2b<0 Thu lai ctung. Vay a = 0, b = 2. Vi du 6: Cho ham sd y = x4 4- 2mx2 - 2 1 0 • 2 (D a) Khao sat va ve do thi (C) khi m = — Duong thang d: y = -6x - 6 nam 2 phia tren hay phia dudi do thi (C). b) Tim quy tich cua diem cue dai cua ham sd (1) Giai 1 '4 2 a) Khi m = — thi ham soy = x + x - 2 2 • Tap xac dinh D = R. Ham so chan . Su bien thien: lim y = +co y' = 4x3 + 2x = 2x(2x2 + 1), y' = 0 co x = 0. BBT X —00 o +0 ° y' 0 y Do thi: y" = 12x + 2 > 0, Vx nen khong cd diem udn. y = 0 co x2 = 1 co x = ±1. Dat y = f(x) = x4 + x2 - 2, y = g(x) = -6x - 6 Xet hieu f(x) - g(x) = x4 + x2 - 2 + 6x + 6 = (x2 - l)(x2 + 2) + 6(x + 1) = (x + l) 2 (x2 - 2x + 4) = (x + l) 2 [(x - l) 2 + 3] > 0, Vx. Vay hai do thi tiep xuc nhau tai diem M(-l ; 0) va do thi (C) d phia tren dudng thang d vdi moi x * -1 . b) y' = 4x3 + 4mx = 4x(x2 + m), y" = 12x2 + 4m Khi m > 0 thi y' = 0 co x = 0, ta cd y"(0) = 4m > 0: diem cue tieu. Khi m = 0 thi khong cd diem cue tri nao. Khi m < 0 thi y' = 0 co x = 0 hoac x = ±V^m , ta cd y"(±V^m) < 0 nen cd 2 diem cue dai: x = ± V-m , y = x4 4- 2mx2 - 2. Khir m thi quy tich cac diem cue dai la dudng cong: y = -x 4 - 2. Vi du 7: Cho ham sd y = x4 - 2mx2 + m3 - m2 (1) a) Khao sat su bien thien va ve do thi ciia ham so khi m = 1. b) Xac dinh m de do thi cua ham sd (1) da cho tiep xuc vdi true hoanh tai hai diem phan biet. -BDHSG DSGT12/1- 19 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Giai a) Khi m = 1 thi y = x4 - 2x2 • Tap xac dinh D = R. Ham sd chin. • Su bien thien lim y = +co. BBT 3 - 4x = 4x(x2 - D, y' = O o x = 0 hoac x = X -co - 1 0 1 +00 y' - 0 + 0 - 0 + y +00 -Y 0 +00 Do thi: y" ' 1 5) 12x2 - 4, y" = 0 co x = ± nen do thi cd 2 diem uon b) y' = 4x3 - 4mx = 4x(x2 - m). De do thi tiep xuc vdi true hoanh tai hai diem phan biet thi dieu kien can va du la phuong trinh y' = 0 cd hai nghiem phan biet khac 0. Neu m < 0 thi x2 - m > 0 vdi moi x nen do thi khong tiep xuc vdi true Ox tai hai diem phan biet. Neu m > 0 thi y' = 0 khi x = 0, x = ±-Jm f(Vm ) = 0 co m2 - 2m2 + m3 - m2 = 0 co m 2 ( m - 2) = 0 co m = 2 (do m > 0) Vay m = 2 la gia tri can tim. 1 5 V i du 8: Cho ham so y = — x4 - 3x2 + - 2 2 a) Khao sat va ve do thi (C) b) Tim a de tiep tuyen cua (C) tai x = a cat (C) tai 2 diem khac nua. Giai a) • Tap xac dinh : D = R Ham sd chan « Su bien thien : lim y = +oo; lim y = +co X—>—X X—> + X y' = 2x3 - 6x = 2x(x2 - 3), y* = 0 co x = 0, x = ± V3 -J3 0 •y \ i ' / * \ ! o / \ / X - i Bang bien thien —00 + 00 + + +00^ 5/2 +00 120 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Ham so dong bien tren (-V3 ; 0), (V3 ;+<*>) nghich bien (-00;-V3 ), (0 ; S ) vaco CD (0; -) ; CT(+ 73 ; -2) . Do thi ddi xung nhau qua true tung: y" = 6x2 - 6, y" = 0<=>X = ±l. Diem udn I(±l; 0) b) Phuong trinh tiep Uiyen tai x = a y = (2a3 - 6a)(x - a) + ^-3a2 +^ = 2a(a2 - 3)x - a 4 + 3a 2 + - 2 2 Phuong trinh hoanh dp giao diem vdi do thi: 3 ) x _ l a ^ + 3 a 2 + - 2 2 l __3 x 2 + ^ = 2a(a 2 2 2 x4 - 6x2 - 4a(a2 - 3)x + 3a4 - 6a (x - a) 2 (x2 + 2ax + 3a2 - 6) = 0 Dieu kien can tim: 0 —v/3 < a < a * ±1 V3" [g(a)*0 Vi du 9: Cho ham so y = x4 - (3m + 5)x2 + (m + l) 2 (1) a) Tim m de do thi ham so (1) cat true hoanh tai 4 diem phan biet cd hoanh do cap so cong. b) Khao sat va ve do thi khi m = 1. Suy ra dd thi y = x4 - 4x3 - 4x2 + 12x - 1. Giai a) Cho y = 0 c=> x4 - (3m + 5)x2 + (m + l) 2 = 0 Dat t = x2 , t > 0 thi PT: t 2 - (3m + 5)t + (m + l) 2 = 0 (2) (3) A = (3m + 5) 2 - 4(m + l) 2 = (5m + 7)(m + 3) Dieu kien (2) cd 4 nghiem phan biet lap cap so cong la (3) co 2 nghiem duong phan biet t,, t 2 (tj < t 2 ): t 2 = 9t x Vi x: = -yjt2 , x2 tt , x3 , x4 = . v /t~vax4 = 3x2 Ta cd t 1 > 2 = -(3 m + 5 ± J(5m + 7) (m + 3)) nen dieu kien: 2 -(3m + 5 +J(5m + 7)(m + 3)=|(3m + 5-V(5m + 7)(m + 3)) -BDHSG DSGT12/1- 121 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
<=> 5V(5m + 7)(m + 3) = 12m + 20 c=> • m _ 19m2 -70m-125 = 0 o m = 5 hoac m = - 25 19 b) Khi m = 1 thi y = x4 - 8x2 + 4 - Tap xac dinh: D = R. Ham so chan • Su bien thien: y' = 4x3 - 16x = 4x(x2 - 4) y' = 0 <=> x = 0 hoac x = ±2. BBT X -co - 2 0 2 +00 y' - 0 + 0 - 0 + y +00 -vr ^12 ' +00 . Dd thi: y" = 12x2 - 16, y' ±2 0 co x = —= nen cd 2 diem udn 73 +2 44 N Ta cd y = x4 - 4x3 - 4x2 + 12x - 1 = (x - l) 4 - 8(x - l) 2 + 6 = f(x - 1) + 2 nen do thi dugc suy ra tir do thi da ve theo phep tinh tien sang phai 1 don vi rdi len tren 2 don vi. V i du 10: Cho ham sd y = x4 + kx2 - k - 1, k la tham sd, do thi la (Ck) a) Khao sat va ve do thi ham so khi k = -1 . b) Chung minh rang do thi (Ck ) luon luon di qua hai diem co dinh A va B khi k thay ddi. Tim k de cho cac tiep tuyen cua (Ck) tai A va tai B vuong gdc vdi nhau. Giai a) Khi k = - 1 , ta cd y = x 4 - x2 • Tap xac dinh D = R. Ham sd chan • Su bien thien: lim y = +oo 72 y' = 4x3 - 2x = 2x(2x2 - 1), y* = 0 co x = 0; x = + — 122 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
BBT: X -oo -V2/2 0 V2/2 +( » y' - 0 + 0 - 0 + y +00 0 +0 ° X y x y -MAT -1/4 Ham so dat CT . Do thi: , CD(0; 0) y y" 12x2 0 <=> x = ± nen do thi cd 2 diem udn 1 5 \ S ' 36 x = 0o y = 0; y = 0 o x2 (x2 - 1) = 0 co x = 0 hoac x = ±1. b) Goi M(x0 ; y0 ) la diem cd djnh: y 0 = x4 + kx2 - k - 1 , Vk o y 0 =k(x:-D+x:-i,vk o 1 = 0 co 1 x =- 1 ,y =0 o ' J 0 X =1 y =0 Vay do thi luon luon di qua hai diem co dinh A(l ; 0) va B(-l ; 0) Ta cd: y' = 4x3 + 2kx. He so gdc tiep tuyen cua (Ck) tai A la y'(l) = 4 + 2k, tai B la y'(-l) = -4 - 2k Dieu kien tiep tuyen nay vuong gdc vdi nhau: y'(l).y'(-D = - 1 co (4 + 2k)(-4 - 2k) = - 1 <=> 4k + 16k + 15 -3 -5 0 co ki = — hoac k2 = — 2 2 C. BA I LUYEN TAP Bai 1: Cho ham sd y = x3 + (1 - 2m)x2 + (2 - m)x + m + 2 (1) a) Khao sat su bien thien va ve do thi cua ham sd (1) khi m = 2 b) Tim cac gia tri cua m de dd thi ham so (1) cd diem cue dai, diem cue tieu, dong thoi hoanh do cua diem cue tieu nhd hon 1. DS: b) - < m < - hoac m < - 1 4 5 -BDHSG DSGT12/1- 123 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Bai 2: Cho ham so y = -x 3 + 3x a) Khao sat va ve do thi b) Viet phuong trinh tiep tuyen cua dd thi ham sd, biet tiep tuyen dd song song vdi dudng thang y = -9x DS: b) cd 2 tiep tuyeny = -9x - 16 vay = -9x + 16. Bai 3: Cho ham sd y = x3 - 3x2 + 2 (D a) Khao sat su bien thien va ve do thi (C) cua ham sd (1) b) Tim tren (C) diem A sao cho khoang each tir A den diem K(2; -4) la nhd nhat. DS: b)A(2;-2) 11 3 a) Khao sat su bien thien va ve do thi (C) cua ham so da cho. b) Tim tren do thi (C) hai diem phan biet M, N doi xung nhau qua true rung. 16" Bai 4: Cho ham so y — + x2 + 3x3 DS: b) M 3; N|-3; f Bai 5: Cho ham so: y = (2 - x)(x + m) 2 a) Khao sat va ve do thi khi m = 1 b) Tim diem co dinh cua do thi ham sd (1) DS: b) diem cd dinh (2; 0) (D m Bai 6: Cho ham so v = —x" 3 (m - l)x 2 + 3(m - 2)x + (D a) Khao sat va ve do thi khi m = 2. b) Tim m de ham sd (1) dong bien tren (2; +oo). DS: b) m > - 3 Bai 7: Cho ham so y = -x 3 + 3x2 - 2, cd dd thi (C) a) Khao sat va ve do thi (C). b) Tim tat ca nhung diem tren dudng thang y duoc 3 tiep tuyen den do thi (C). 2 ma tir do cd the ke DS: b) a < - 1 hay a > — 3 a * 2 Bai 8: Cho ham sd y = x3 + mx2 - x - m (1) cd do thi (C m ) 124 : -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
a) Khao sat ham s6 (1) voi m = 1. b) Tim m de (C m ) cit true hoanh tai ba diem phan biet va hoanh do cac giao diem lap thanh mot cap sd cong. DS:b)m = 0; + 3 Bai 9: Cho ham so y = x(4x2 + m) a) Khao sat sir bien thien va ve do thi ham so khi m = -3 b) Tim m de I y I < 1 vdi moi x e [0; 1] DS: b)m = -3 Bai 10: a) Khao sat va ve do thi (C) ciia ham sd y = — - 2(x2 -1) b) Viit phuong trinh cac dudng thang di qua diem A(0; 2) va tiep xuc vdi (C). DS: b) d2,3:y = ±^J|x + 2 Bai 11: a) Khao sat va ve do thi (C) ciia ham so y = x4 - 6x2 + 5 b) Tim m dk phuong trinh x 4 - 6x2 - m = 0 cd 4 nghiem phan biet. DS: b) -9<m< 0 Bai 12: Cho ham so y = x4 + 2(m + l)x 2 + 1 a) Khao sat va ve do thi ham so khi m = 1. b) Tim m de dd thi ham sd cd 3 diem cue tri. Tim phuong trinh dudng cong di qua cac diem cue tri dd. DS: b) y = (m + l)x 2 + 1 Bai 13: Cho ham so y = 2mx4 - x2 - 4m + 1 (1) a) Khao sat va ve do thi khi m = -1 . b) Tim m dk db thi ham sd (1) cd 2 cue tieu va khoang each giua chung bang 5. DS: b) m = — 25 -BDHSG DSGT12/1- 125 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
§ 6 . KHA O SA T V A V E HA M HU U T I A. KIE N THU C CO BAN So- 36 chung ve khao sat va ve do thi ham huu ti : Gdm 3 budc: Budcl: Tap xac dinh - Tim tap xac dinh - Xet tinh chan, le neu cd. Budc 2: Chieu bien thien - Tinh cac gidi han. - Tim cac tiem can - Tinh dao ham cap mot, xet dau - Lap bang bien thien rdi chi ra khoang ddng bien, nghich bien va cue dai, cue tieu. Budc 3: Ve do thi - Cho vai gia tri dac biet, giao diem vdi hai true toa do. - Ve dung do thi, luu y tam doi xung la giao diem 2 tiem can. B. PHAN DANG TOA N DANG 1: HAM s 6 y . ax + b cx + d (c*0vaad-bc*0). x 3.X ~f" b Cac dang do thi ham huu ti 1/1: y = vdi c * 0, ad - be * 0 cx + d r V i du 1: Cho ham sd y 2x- l x - l a) Khao sat su bien thien va ve dd thi (C). b) Tim cac diem tren (C) cd toa do la sd nguyen. Giai a) • Tap xac dinh D = R|{ 1} . Su bien thien: lim y = -oo, lim y = +co nen tiem can dungx-»r x->r ' ° x = 1. Ta cd lim y = 2 nen tiem can ngang: y = 2. - 1 (x-l ) < 0, Vx * 1. Ham so khdng cd cue tri. 126 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
BBT X -00 1 +00 y' + y 2 -00 +00 2 Ham so nghich bien tren moi khoang (-oo; 1) va (1; +oo) . D6 thi: Cho x = 0=>y=l; y = 0=>x= - Do thi nhan giao diem 1(1; 2) cua hai dudng tiem can lam tam ddi xung. b) y 2x- l = 2 + 1 x - l x- l Diem M(x; y) e (C) cd toa do nguyen khi x - 1 = ±1. Suy ra (C) cd 2 diem (0; 1) va (2; 3) cd toa do la so nguyen. - x + 2 Vi du 2: Cho ham sd y x + 2 a) Khao sat su bien thien va ve dd thi (C) cua ham sd. b) Viet phuong trinh tiip tuyen cua (C), biet nd vuong goc vc 1 thang y = - x - 8. 2 a) . Tap xac dinh: D = R \ {-2} • Su bien thien: Tiem can dung x = -2 vi lim y = +oo, lim y = -oo. t->(2) x->(2)* Tiem can ngang y = - 1 vi lim y = -1 ; x->±x -4 y (x+2) 2 Bang bien thien < 0 , Vx^-2 . X -oo _2 +00 y' y -1 \ * -CO +00 -1 Ham sd nghich bien tren cac khoang (-oo; -2) va (-2; +oo). . Dd thi: Cho x = 0=>y=l; y = 0=>x = 2. -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
b) Tiep tuyen vuong goc vbi dubng thing y = — x - 8 nen he sd goc k — 2 . 2 Hoanh do tiep diem thoa man phuong trinh: -4 = -2 => x, =- 2 + J2 (x + 2) 2 (x 2 =-2-V 2 Vdi Xj = -2 + 72 , ta cd tiip tuyen y = -2x - 5 + 4 %/2 Vdi x2 = -2 - s/2 , ta cd tiep tuyen y = -2x + 5 - 4 V2 x - 3 V i du 3: Cho ham sd y = . , 2 ~ x a) Khao sat va ve do thi (C) cua ham so. b) Chung minh do thi (C) cd tam doi xung. Giai a) . Tap xac dinh D = R \ {2} • Su bien thien lim y = -co, lim y = +00 nen dudng thang x = 2 la tiem can dung. x->2~ x->2* lim y = - 1 nen dudng thang y = 1 la tiem can ngang v—*+cf y - - 1 < 0, Vx e R \ {2} : Ham so khong cd cue tri, ham so nghich (2-x) 2 bien tren moi khoang (-co: 2) va (2; +co) Bang bien thien X -00 2 ~4 ~°° y' y -1 ^^*-co +CC -1 • Do thi: Chox = 0 y - - ; y = o 2 J b) Giao diem cua hai tiem can la 1(2; -1). Ap dung cong thuc chuyen he bang phep tinh tien vecto fx = X + 2 ly = Y- i D6 thi (C) trong he toa do IXY: O I : Y - 1 Vi Y = F(X) (X + 2)- 3 2-( X + 2) 1 <=> Y : : X X la ham le nen do thi nhan goc I la tam doi xung. L 0 -1 2 3 w X -1,5 • \ 1 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Vi du 4: Cho ham sd y x + 1 a) Khao sat su bien thien va ve do thi (C) cua ham so. x - 2 b) Bien luan theo m so nghiem cua phuong trinh 1 r x +1 2 m+ 1. Giai a) . Tap xac dinh D = R \ {-1} . o . Chieu bien thien: y' (x + 1) mdi khoang (-co; -1) va (-1 ; +co) Tiem can dung x = -1 ; Tiem can ngang y = 1. Bang bien thien > 0, Vx ^ - 1 nen ham so dong bien tren X -CO _1 +CO y' + + y +CO 1 / / 1 -CO • Do thi: Cho x = 0 => y = -2. Cho y = 0 => x x - 2 J yi l ° » -1 - I X \ -2 V = 2m+l 2. \ b) Sd nghiem cua phuong trinh x + 1 2m + 1 la sd giao diem cua do thi (C) cua ham sd y T . x- 2 la co y = -. r = •( x + 1 |x + l | x - 2 x + 1 x - 2 vdi dudng thang y = 2m + 1. khi x >-1 khix < -1 x + 1 Suy ra do thi (C) giu nguyen phan do thi (C) nam ben phai dudng thang x = - 1 va lay doi xung phan ben trai dudng thang x = - 1 qua true hoanh. Dua vao do thi ta cd: Neu 2m + 1 < - 1 c=> m < - 1 thi phuong trinh cd 2 nghiem phan biet. Neu - 1 < 2m + 1 < 1 o - 1 < m < 0 thi phuong trinh cd 1 nghiem duy nhat. Neu 2m + 1 > 1 o m > 0 thi phuong trinh vd nghiem. 4 Vi du 5: Cho ham so y = (1) 2 - mx a) Khao sat va ve do thi khi m = 1. b) Tim cac diem cd dinh cua dd thi (1) va cac diem ma cac do thi (1) khdng di qua vdi moi m. -BDHSG DSGT12/1- 129 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Giai a) Khi m = 1 thi y = 2 - x Tap xac dinh: D = R\ {2}. Su bien thien: y' = > 0, Vx * 2 nen ham sd ddng bien tren tirng (2-x) 2 khoang xac dinh (-co; 2) va (2; +<x>). Ham sd khong cd cue tri. Dirdng thang x = 2 la tiem can dung, vi lim y = +co, lim y = -co x->2~ x-»2* Dudng thang y = 0 (true hoanh) la tiem can ngang vi lim y = lim 2 - x = 0. Bang bien thien X -co 2 +co y1 + + y +CO 0 ^ -co • Do thi: Do thi cat true tung tai diem A(0; 2). b) Gpi M(x0 ; y0 ) la diim cd dinh cua do thi (1): 4 fx =0 y„= - . Vm o ' 2-mx o [y o = 2 Vay cac do thi (1) ludn luon qua diem co dinh M(0; 2). Gpi N(x0 ; y0 ) la diem ma cac do thi (1) khdng di qua: 4 fx =0 y0 * Vm <=> <^ ° 2 - m x o [Vo* 2 Vay tap hpp cac diem ma cac dd thi (1) khong di qua la dudng thang x = 0 (true tung) trir diem co dinh M(0; 2). V i du 6: Cho ham so y = 3 x + 1 , x + 1 a) Khao sat su bien thien va ve dd thi cua ham sd. b) Tinh dien tich cua tam giac tao bdi cac true toa dp va tiep tuyln cua do thi ham sd tai diem M(-2; 5). Giai a) • Tap xac dinh D = R \ {-1} 2 . Su bien thien y' = — > 0, Vx e D nen ham sd ddng bien tren (-oo; -1) (x +1) va (-1 ; +oo). Tiem can dung: x = -1 , tiem can ngang y = 3. 130 -BDHSG DSGTllllDownload Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Bang bien thien X -00 . ] +co y' + + y +GO 3 ^ ^ - 3 -00 . Do thi: Cho x = 0 Cho y = 0 y = l . x = Tam doi xung I(-l ; 3) b) Phuong trinh tiep tuyen d cua do thi ham so tai M la: y = y'(-2)(x + 2) + 5 co y = 2(x + 2) + 5 eo y = 2x + 9. 9 Duong thang d cat true hoanh tai ; 0) va cat true tung tai B(0; 9). Dien tich tam giac OAB la S = - OA.OB = - 2 2 9 nl 8 1 (A A*\ —;9 =— (dvdt). 2. 1 4 Vi du 7: Cho ham so y = x + 2 x - 3 a) Khao sat su bien thien va ve do thi (C) cua ham so. Chung minh giao diem I cua hai tiem can cua (C) la tam doi xung cua (C). b) Tim diem M tren do thi cua ham so sao cho khoang each tir M den tiem can dirng bang khoang each tir M den tiem can ngang. Giai a). Tap xac dinh: D = R \ {3} . -5 (x-3) 2 < • Su bien thien: y' (-oo; 3), (3; +w). lim y = -co, lim y = +co nen TCD: x = 3. x->3~ x-»3* lim y = 1 nen TCN: y = 1. x->±r Bang bien thien 0, Vx 3 nen ham sd nghich bien tren X -co 3 +00 y' y 1 -00 +00 . Dd thi: -BDHSG DSGT12/1- y i 1 I V O -2/3 . 3 x 131 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Cho x = 0=>y = — , y = 0 => x = -2 3 Giao diem 2 tiem can 1(3; 1). Chuyen true bang phep tinh tien vecto Onr.f I x = X + 3 Y + l Dd thi (C): Y+ l = (X+3)+ 2 <=> Y = — la ham sd le (X+3)-3 X dpcm. b) Gia su M(x0 ; y0 ) e (C). Goi di la khoang each tir M den tiem can dung va &2 la khoang each tir M den tiem can ngang thi: i i i i 5 dj = | x0 - 31, d2 = I y0— 1 x. - 3 Ta cd xn - 3 = o x0 -= 3 +4b Vay M(3 S M'(3 + VD ; 1 + S ) . 3-2 x V i du 8: Cho ham sd y x - l a) Khao sat su bien thien va ve dd thi (C) cua ham sd. Suy ra do thi (C): 3-2 x x - l b) Tim tat ca cac gia tri cua tham sd m de dudng thang y = mx + 2 cat do thi cua ham sd tai hai diem phan biet. Giai a) . Tap xac dinh D = R \ {1}. . Su bien thien: y' = -—- < 0, Vx e D nen ham sd nghich bien tren (x-l ) mdi khoang (-co; 1) va (1; +co). lim y = +co va lim y = -co nen TCD: x = 1 X->1* x-t-1 lim y = lim y = -2 nen TCN: y = -2. Bang bien thien X -CO 1 +CO y' y -2 -CO +00 -2 Dd thi: Cat true tung tai diem (0; -3) va true hoanh tai diem ( - 0) 2 132 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
y | 0 1 \3/2 -2 X i I y - 3 / 1 2 \ r . 0 1 3/2 x Ta co y 3-2 x 1 3-2 x 3 khi 1 < x < — x - l 2 3-2 x 1 U , 3 kW x < 1, x > - x - l 2 nen do thi (C) giu nguyen phan dd thi (C) d phia tren Ox, con phan dudi Ox lay ddi xung qua Ox. b) Duong thang y = mx + 2 cat dd thi tai hai diem phan biet. <=> Phuong trinh 3-2 x mx + 2 cd hai nghiem phan biet. <=> Phuong trinh mx2 - (m - 4)x - 5 = 0 cd hai nghiem phan biet, khac 1. m * 0 A > 0 m * 0 g(D*0 m2 + 12m + 16>0 m < -6 < 2V5 m > -b + ,m * 0 Vi du 9: Cho ham sd y x + 2 (1) 2x + 3 a) Khao sat su bien thien va ve dd thi cua ham sd (1) b) Viet phuong trinh tiep tuyen cua dd thi ham sd (1), biet tiep tuyen 6 5 cat true hoanh, true tung lan luot tai hai diem phan biet A, B va tain giac OAB can tai gdc toa do O. Giai 3 a) . Tap xac dinh D = R \ {-—} • Su bien thien: y' = —=• < 0, Vx e D nen ham sd nghich bien tren (2x + 3) 2 ' 3 3 ' . (-00: — )va( — ; +00). Ham sd khdng cd cue tri 2 2 1 - - 1 hm y = hm y = - nen tiem can ngang: y = - 3 lim y =-co, lim y = +co nen tiem can dung: x = — rj -BDHSG DSGT12/1- 133 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Bang bien thien X -CO -3/2 +00 y' y 1/2 \ * -co +00 ^ 1/2 . Do thi: Cho x = 0 => y = - 3 y = 0 =>x = -2 . b) Tam giac OAB vuong can tai O, suy ra he so goc tiep tuyen bang ±1. Goi - 1 y V i 2/3 ——— ^ 1/2 A . •3/2 0 x toa clo tiep diem la (x0 ; y0 ), ta cd = ±1 (2x o+3) 2 co x0 = - 2 hoac x 0 = -1 . Vdi x 0 = -1 , y 0 = 1 thi phuong trinh tiep tuyen y = - x (loai) vi A, B triing nhau tai O. Vdi XQ = -2 , y 0 = 0 thi phuong trinh tiep tuyen y = - x - 2 (thoa man). Vay, tiep tuyen can tim: y = - x - 2. 2x +1 V i du 10: Cho ham sd y = , x ~ 2 a) Khao sat va ve dd thi (C) b) Tim cac diem tren dudng thang d: x = 3 ma tii do ve duoc tiep tuyen den do thi (C). Giai a) - Tap xac dinh D = R \ {2} . • Su bien thien: lim y = 2 nen TCN: y = 2. lim y = -co, lim y = +oo nen TCD: x = 2. x->2~ x->2* . Ta cd y' = ~~5 , < 0, Vx e D i ( x -2) 2 khoang xac dinh (-co; 2) va (2; +oo). BBT: X -CO 2 +00 y' y 2 \ ^-00 +00 N ^ 2 134 . Dd thi: Tam ddi xung la giao diem 2 tiem can 1(2; 2). ham sd n ghich bien tren rirnj > 1 v 2 2 0 -1/2 2 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Cho x = 0=> y = --, y = 0=>x = - - 2 2 b) Goi M(3; b) e d. Phuong trinh tiep tuyen qua M he so gdc k: y = k(x-3 ) + b. Ta tim dieu kien he sau cd nghiem x: '2x + l f(x) = g(x) f(x ) = g, (x) x - 2 . -5 l(x-2) 2 (x-3 ) + b = k(x-3 ) + b = k Dodd ^ ± l = -^ - x - 2 (x-2)' o (b - 2)x2 - 2(2b .+ l)x + 4b + 17 = 0, x * 2. 5 Xet b = 2 thi he cd nghiem x = — (chon) Xet b *2 thi dieu kien A' > 0, y(2) * 0 <=> b < 7. Vay cac diem can tim M(3; b) vdi b < 7. DANG 2:HAMS6y : ax +bx + c a'x + b' (a * 0, a' * 0) Cac dang do thi ham hiru ti :y - a x + bx + c ^ ^ Q a i ^ a'x + b' Vi du 1: Cho ham sd y x2 + 2x + 5 x + 1 a) Khao sat su bien thien va ve dd thi (C) cua ham sd. b) Tim m de phuong trinh sau cd hai nghiem duong phan biet: x2 + 2x + 5 = (m2 + 2m + 5)(x + 1). Giai , x2 +2x + 5 , , , 4 a) y = • = x + 1 + n x+ 1 x+ 1 • Tap xac dinhD = R\ {-1} . 4 . Su bien thien: y' = 1 - x'+2x- 3 , . — . y' = 0ox = l, x = -3. x + 1 (x + 1) •BDHSG DSGT1Z/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Bang bien thien X -CO -3 -\ 1 +00 y' + 0 - - 0 + y -CO -00 +00 +oo 4 Ham so dbng bien tren (-co; -3) va (1; +oo), nghich bien tren (-3; -1) va (-1 ; i). Ham sd dat CD (-3; -4), CT(1; 4). Ta cd lim y = -co, lim y = +<x> I-K-D" x-»(-n* nen TCD: x = 2 lim (y - (x +1)) = hm -4 — = 0 X->±co X->±eO X + 1 nen TCX: y = x + 1. . Dd thi: Cho x = 0 => y = 5 Tam doi xung la giao diem 2 tiem can I(— 1; 0). b) V i x = - 1 khong la nghiem nen phuong trinh da cho tuong duong vdi: x2 + 2x + 5 \ i i y 4 i y -3 I 1 <S 1 -1 3 1 x -4 X + 1 m + 2m + 5. Sd nghiem cua phuong trinh bang so giao x2 + 2x + 5 < 2 diem cua dd thi ham sd y = vdi dudng thang y = m + 2m + 5. x + 1 Phuong trinh cd hai nghiem duong khi va chi khi: fm * - 1 4 < m 2 + 2m + 5<5<=> -2 < m < 0 V i du 2: Cho ham sd y x2 +4 a) Khao sat su bien thien va ve dd thi (C) cua ham sd. b) Tim m sao cho dudng thang y = m(x - 2) + 4 cat dudng cong (C) tai hai diem thudc hai nhanh cua nd. Giai 4 a) .Tap xac dinh D = R \ {0}. Ta cd y = x + — la ham sd le. X . Su bien thien: lim y = -oo, lim y = +co nen dudng thang x = 0 la tiem can dung X-+CT x->0* 4 lim (y-x) = lim — nen duong thang y = x la tiem can xien. x->±w x->±^ X 136 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Ta co: y' = 1 - BBT 4 _x 2 - 4 x2 " x2 ;y' = 0o x = ±2. X -oo -2 0 2 +°o y' + 0 0 + y -4 -CO -00 +CO +00 \ y 4 Ham so dong bien tren moi khoang (-oo; -2) va (2; +co), nghich bien tren mdi khoang (-2; 0) va (0; 2). Ham sd dat cue dai tai diem x = -2,y CD = -4. i Ham sd dat cue tieu tai diem x = 2, y C T = 4. . Dd thi: b) Hoanh do giao diem cua dudng thang va dudng cong (C) la nghiem cua 4 phuong trinh: m(x -2 ) + 4 = x + —co mx(x - 2) + 4x = x2 + 4, x * 0. x CO (m - l)x2 - 2(m - 2)x - 4 = 0, x * 0. Hai nhanh cua (C) nam ve hai ben cua dudng tiem can dung x = 0. Duong thang da cho cat (C) tai hai diem thudc hai nhanh cua nd khi va chi khi phuong trinh (1) cd hai nghiem trai dau. m-l* 0 _4 com-l>0com>l . <0 fa* 0 P< 0 2x- 3 m + 1 Vi du 3: Cho ham sd y X — z a) Khao sat su bien thien va ve do thi (C) cua ham sd. Tim cac diem tren (C) cd toa do la sd nguyen. b) Chung minh dd thi (C) cd tam ddi xung. Giai 3 a) Ta cd y = x x - 2 Tap xac djnhD = R\ {2}. . Su bien thien : lim y = +co va lim y = -co nen TCD: x = 2. x->2 x->2* lim (y - x) = lim X->±X X ' = 0 nen TCX: y = x. -BDHSG DSGT12/1- 137 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
1 + (x-2) 2 khoang (-co; 2) va (2; +oo). Bang bien thien: > 0 vbi moi x * 2 nen ham so ddng bien tren moi X -co 2 +oo y' + + y +00 y -00 +00 y -00 . Do thi: Cho x = 0 => y = - 2 y = 0 = > x = -l, x = 3. Diem M(x; y) e (C) co toa do nguyen khi x - 2 la udc sd cua cua3 nenx - 2 = ±1 , ±3. Do dd (C) cd 4 diem cd toa do nguyen:(l; 4), (3; 0), (-1 ; 0) va (5; 4). b) Giao diem 2 tiem can 1(2; 2) chuyln true bang phep tinh tien vecto OI:j X = X + 2 [y = Y + 2 Do thi (C): Y + 2 = (X + 2) - 3 (X + 2)- 2 ^ Y X X V i Y = F(X): X tam doi xung. X la ham so le nen dd thi (C) nhan gdc 1(2; 2) lam V i du 4: a) Khao sat va ve do thi (C) cua ham so: y 1-x 2 b) Tim toa dd diem M d tren do thi (C) sao cho tiep tuyln cua dd thi (C) tai diein M cat hai tiem can cua (C) tai hai diim A, B va dd dai AB ngan nhat. Giai a) y 1-x2 1 = - x + — X X .Tap xac dinh : D = R \ {0} . Ham sd le • Su bien thien : TCD: x = 0, TCX: y = - x 1 -1 - < 0, V x e D 138 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
Bang bien thien y +00 +00 +00 -00 . 0 6 thi y = 0o x = ± l b) Phuong trinh tiep tuyen tai M0 (x0 , yo) (A):y = - 1 - 1 1 "•o J (x-x 0 )-x 0 + 1 Toa do giao diem cua (A) vdi TCD, TCX: A(0; — );B(2x0; -2x0) AB2 = 8 x2 + — + 8 > 8%/2 + 8 Dau = khi 8x2 = — co x4 , = — co x„ = ±—= 0 x2 Vay diem can tim: M Vi du 5: Cho ham so y 1 • 1 +V5 V2 M ' V2 72 J x - l * x a) Khao sat va ve do thi (C) cua ham so b) Tim m de dudng thang y = 3x + m cat do thi (C) tai 2 diem phan biet co hoanh do xi , X; va | xx - x2 1 dat gia tri nhd nhat. Giai 1 a) Ta cd: y -=x + 1 + x - l x- l • Tap xac dinhD = R \ {1}. i • Su bien thien y' = 1 x2 -2 x (x-l) 2 (x-l) 2 y' = 0 o x = 0 hoac x = 2. x = 1 la tiem can dung vi lim y = -co, lim y = +co y = x+1 la tiem can xien vi lim (y - (x+1)) = lim X—>±co X—>±t» X — ]_ BBT X -00 0 i 2 +00 y' + 0 0 + y -00 0 -001 +00 \ / 4 +00 -BDHSG DSGT12/1- 139 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
. 0 6 thi: b) Phucmg trinh hoanh do giao diem: -= 3x + m x - l c=> 2x2 + (m - 3)x - m = 0, x * 1. Dieu kien cd 2 nghiem phan biet khac 1 JA> O U d ) * o Ta cd: x : - x2 = V m2 + 2m + 9 1 r -T— - ^2 = —v(m + l) +8 > — 4 4 Vay gia tri | xi - X21 nhd nhat khi m = - 1 V i du 6: Cho ham so y x2 -3 x + 3 x - l a) Khao sat va ve do thi. b) Chung minh rang qua diem M(3; -1) ve duoc hai tiep tuyen vdi do thi va hai tiep tuyen do vuong gdc vdi nhau. Giai 1 a) • Tap xac dinh D = R\ {l}, y = x - 2 + x - l x 2 - 2 x • Su bien thien y' = . y' = 0 khi x = 0 hoac x = 2. (x-l) 2 TCD: Bang x = 1,TCX bien thien y = x - 2 . X -00 0 1 2 +oo y* + 0 0 + y -3 -00 -co| +00 +00 1 . 06 thi x = 0 thi y = -3 Tam doi xung 1(1; -1) b) Phuong trinh duong thang qua M(3; -1) he s6 gdc a la y = a(x - 3) - 1. dudng thang la tiep tuyen vdi do thi khi he sau cd nghiem: 140 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
[f(x) = g(x) [f'(x) = g, (x) 3x + 3 x - l - 2 x a(x-3)- l (1) (2) .(x-l) 2 Thay (2) vao (1) va nit gon ta duoc: x2 - x - 1 = 0. PT cd 2 nghiem thoa man: x, + x2 = l,xi.x 2 = -1 . Taed: y'(x1).y'(x2) = -7±=^L ( x t -l) 2 (x 2 -l) 2 (x^) 2 -2x 1 x 2 ( x 1 +x2 ) + 4x1 x2 _ 1 + 2- 4 (x^-x^Xj+l) 2 Vay 2 tiep tuyen qua M vudng gdc voi nhau. (-1-1 + 1)2 Vi du 7: Cho ham sd y = x 2 + l a) Khao sat va ve dd thi ham sd. Tinh gdc giua 2 tiem can. b) Bien luan theo m sd nghiem cua PT: Giai x 2 + l m 2 + l in a) Ham sd y = x + 1 • Tap xac dinh D = R \ {0} . Ham sd le. ..2 1 Su bien thien : y '= •,y' = 0co x l hoac x = l . lim y = -co, lim y = +oo nen TCD: x = 0 x-»o x->o+ lim (y - x) = lim — = 0 nen TCX: y = x. X —> ±« x —> ±C0 Bang bien thien X -00 - l 0 l +00 y' + 0 0 + y -00 .9 -GO +00 \ 2 +00 . Dd thi: Ddi xung nhau qua gdc O. TCD: x = 0, TCX: y = x nen hai tiem can hop nhau goc 45° b) Sd nghiem phuong trinh x2 + 1 m2 +1 m -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
la so giao diem cua do thi vdi dudng thang y n r +1 m f(m). Dua vao dd thi ta cd: J = f(m) Neu mz +1 < -2 hoac m' +1 > 2 m m o m # 0, m * ±1 , thi PT cd 2 nghiem Neu m2 + 1 -2 hoac mz +1 = 2 m m co m = - 1 hoac m = 1 thi PT cd 1 nghiem. Cdn khi m = 0 thi PT vd nghiem. YJ du 8: Cho ham sd y = x + 2x cd do thi (C) X 1 a) Khao sat va ve dd thi (C). Suy ra do thi y : x2 +2x x - l b) Viet phuong trinh dudng thang di qua hai diem cue tri cua (C). Giai a) Tap xac dinh D = R\ {l}.Tacd y = x + 3 x - l • Su bien thien: 3 y' = l - x -2x- 2 (x-l) 2 (x-l) 2 , y' = 0ox = l ± V 3 lim y = -GO. lim y = +co nen TCD: x = 1 x->l* lim (y - (x + 3)) = lim = o nen TCX: y = x + 3 Bang bien thien: X -co \-S 0 1+V3 +co y' + 0 0 + y 4-2V3 -00 -00 +00 +00 4-2V3 1 Dd thi: y = 0 <=> x = 0 hoac x = -2. 142 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
khi - 2 < x < 0 hay x > 1 Ta co y x - l khi x - 2 hay 0 < x < 1 nen co do thi gid nguyen do thi (C) phan phia tren Ox va lay ddi xung phan phia dudi Ox qua Ox. b) Dd thi cua ham sd da cho cd hai diem cue tri la: A (l - 73 ; 4 -2V 3 )va B(l + 73 ; 4 + 2 73 ). Duong thang di qua hai diem cue trj cua dd thi la dudng thang d cd vecto chi phuong u = — ^ AB = (1; 2) va di qua diem A nen cd phuong trinh: 273 x-(l-73 ) y-(4-273 ) 1 Vi du 9: Cho ham sd y co2x-y + 2 = 0. mx -m x + 1 x - l (1) a) Tim diem cd dinh cua dd thi ham sd (1). b) Khao sat va ve dd thi (C) khi m = 1. Suy ra dd thi ham sd y Giai a) Goi M(x0 ; y0 ) la diem cd dinh ciia dd thi (1): m x 2 - m x +1 w m(x 2 - x j + l w y0 = 2 _2 . V/m <=> y 0 = ——2 °- . Vm x2 - X + 1 X - 1 CO xo - 1 x 2 - x o = 0 , x o - l *0 1 <=> y0 x - 1 o x =0 y„ x 0 - l Vay cac dd thi ludn ludn qua M(0; -1). -BDHSG DSGT12/1- - i 143 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
b) Khi m = 1 thi y x - x + 1 1 x - l . Tap xac dinh D = R \ {1} • Sir bien thien y' = 1 - x + - x - l x2 -2x (x-l) 2 " (x-l) 2 1 , y' = Oo x = 0, x = 2. Bang bien thien: X -oo 0 1 2+o o y' + 0 0 + y -1 -oo -coj + 0C +00 3 . Do thi Ta cd y x2 - X + 1 X - 1 la ham sd chan nen dd thi (C) ddi ximg nhau qua Oy. Khi x > 0 thi lay phan dd thi (C), sau do lay ddi xung phan dd qua Oy thi duoc dd thi (C). x2 +(m-l) x + 2 V i du 10: Cho ham sd y 1-x a) Xac dinh m de ham sd dat cue tri tai xi, x2 sao cho XjX2 = -3. b) Khao sat va ve dd thi ham sd tren khi m = 2. Giai „ „ . „ , -x 2 + 2 x + m + l a) D = R\{l}.Tacoy' = — y' = o co x2 - 2x - m - 1 = 0, x * 1 (A' = 2 + m) f2 + m> 0 Ham sd dat cue tri tai xi , x2 va XjX2 = -3 co \ co m = 2. -m- l = -3 b) Khim = 2thi y x2 + x + 2 „ 4 -x- 2 1-x .Tap xac dinhD = R\ {1}. x - l 144 -BDHSG DSGTUIV Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
-x2 + 2x + 3 • Su bien thien. y' = - 1 + = (x-l) 2 (x-l) 2 y' = Oo x = - l hoac x = 3. lim y = +oo, lim y = -oo nen TCD: x = 1 x—»1 x—>1 + lim (y - (-x - 2) = lim —— x—y±°c x—*±x. x 1 Bang bien thien 0 nen TCX: y = - x - 2. X -00 -1 3 +oo y' 0 + + 0 y +00 +0C -co -OC • Do thi: Cho x = 0 => y = 2. Tam doi xung 1(1; -3). C. BAI LUYEN TAP Bai 1: a) Khao sat va ve do thi cua ham so: y x - l (D b) Chung minh rang vdi moi m * 0 thi dudng thang y = mx - 2m luon luon cat do thi (1) tai 2 diem phan biet va trong do co it nhat mpt giao diem cua nd cd hoanh dp duong. 2x- 4 Bai 2: a) Khao sat va ve dd thi (C) cua ham sd y = x + 1 b) Chung minh rang khdng cd tiep tuyen nao cua do thi (C) di qua giao diem cua hai tiem can cua dd thi do. HD: b) dung phan chung Bai 3: Cho ham sd y = X + ^ 3 x- l a) Khao sat su bien thien va ve dd thi (C) cua ham sd da cho. b) Cho diem M0 ( x 0 ; yo) thudc dd thi (C). Tiep tuyen cua (C) tai M 0 cat cac tiem can cua (C) tai cac diem A va B. Chung minh M 0 la trung diem cua doan thang AB. _ , x x - m Bai 4: Cho ham so y = — (vdi m la tham sd) (D a) Khao sat su bien thien va ve dd thi cua ham sd (1) khi m = 1. -BDHSG DSGT12/1- 145 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
b) Tim cac gia tri cua m de tren dd thi cua ham so (1) cd it nhat mpt diem each deu hai true toa dp, ddng thdi hoanh dp va tung dd cua diem nay trai dau nhau. DS: b) m > - i 4 x + 3 Bai 5: Cho ham sd y = (*) x + 2 a) Khao sat su bien thien va ve dd thi cua ham sd (*) b) Gpi (C) la do thi cua ham sd (*) da cho. Chung minh rang dudng thang y = — x - m ludn cat (C) tai hai diem phan biet A va B. Xac dinh m sao cho dp dai doan AB la nhd nhat. DS: b) m = -2 nu u- * x 2 - 2 x + 2m- l Bai 6: Cho ham so y = (C m ) x - l 3 a) Khao sat va ve dd thi ham sd khi m = — 2 b) Tim m sao cho ham sd cd cue dai, cue tieu va khoang each giua diem cue dai va diem cue tieu cua dd thi ham sd bang 5. DS: b) m = — 8 Bai 7: Cho ham sd y = x + 5x + 5 x + 1 a) Khao sat su bien thien va ve dd thi (C) cua ham sd (1). b) Tim m de dudng thang y = m cat dd thi (C) tai M , N (x M < x N ); ddng thdi cat tiem can dung, tiem can xien lan lyot tai P, Q sao cho PM = NQ. DS: b) m < 1 hoac m > 5 ( x -l) 2 Bai 8: Cho ham sd y = ^ cd dd thi (C) x - 2 a) Khao sat su bien thien va ve dd thi (C) b) Xac dinh ham sd y = f(x) cd dd thi ddi xung vdi (C) qua A(l ; 1) DS: b) y = f(x) = x + - x x2 — 4x + 3 Bai 9: a) Khao sat va ve dd thi (C) cua ham sd y = x - 2 b) Tim cac gia tri ciia m d§ phuong trinh: | x2 - 4x + 3 | = m(x - 2) cd 2 nghi?m duong-phan biet 146 • -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
DS: b) m>- - 2 Bai 10: a) Khao sat ham so: y = x + 2 x + 2 ^ x + 1 b) Dung dd thi ciia ham so (1), bien luan theo a so nghiem ciia phuong x 2 + 2 x + 2 a 2 +2 a + 2 trinh: x + 1 a +1 2x2 + x + 1 Bai 11: Cho ham so: y = ±zz_L±Z± (C) x + 1 a) Khao sat va ve do thi (C) b) Chung minh rang tich cac khoang each tir mot diem M bat ki tren do thi (C) den hai dudng tiem can cua no ludn la mdt hang sd. 2 DS: b) dj.dz = —= hang sd. v5 x2 - x- l Bai 12: Cho ham sd y x + 1 a) Khao sat su bien thien va ve dd thi (C) cua ham sd da cho. b) Viet phuong trinh cac tiep tuyen cua (C) di qua diem A(0; -5). DS: b) (A,):y = -5,(A 2 ) : y = -8x- 5 Bai 13: Cho ham sd f(x) = 2x2 ~ m x + m + 6 (Cm ) x - l a) Khao sat va ve dd thi (Ci) khi m = 1. b) Dinh m de (C m ) cat Ox tai 2 diem phan biet ma tiep tuyen tai dd vudng gdc vdi nhau. DS: b) m = 4 ± 2Vl7 x2 — 3x Bai 14: Cho ham sd: y = (m la tham sd) x - m a) Khao sat ham sd (1) khi m = -1 . b) Tim m de ham sd (1) ddng bien tren [1; +oo). DS: b) - 1 < m < 1. -BDHSG DSGT12/1- 147 Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com
§ 7 . BA I TOA N THTJYJNG GA P V E D O TH I - Bai toan ve ddng bien, nghich bien - Bai toan ve diem eye dai, cue tieu. diem udn - Bai toan ve tuong giao, giao diem - Bai toan ve tiep tuyen, tiep xuc - Bai toan ve yeu td ddi xung, ve tap hop diem - Bai toan ve diem cd dinh, diem ho khdng di qua - Bai toan ve gdc, khoang each - Bai toan ve tiem can - Bai toan ve gia tri ldn nhat, nhd nhat - Bai toan ve xac dinh ham da thuc, ham huu ti: Ta dua vao cac gia thiet cho de lap phuong trinh, he phuong trinh, tir do tim ra cac he sd cua ham sd. - Bai toan ve suy dd thi vdi phep ddi xung: Tir do thi (C): y = f(x) suy ra cac dd thi: y = - f(x) bang each lay ddi xung qua true hoanh. y = f(-x ) bang each lay ddi xung qua true tung. y = - f(-x ) bang each lay ddi xung qua gdc. y = | f(x)| bang each lay phan dd thi d phia tren true hoanh, cdn phan phia dudi true hoanh thi ddi xung qua true hoanh. y = f(|x|) la ham sd chan, bang each lay phan do thi d phia ben phai true tung, rdi lay ddi xung phan do qua true tung. - Bai toan ve suy dd thi vdi phep tinh tien: Tir do thi (C): y = f(x) suy ra cac dd thi: y = f(x) + p, p > 0 bang each tinh tien len tren p don vi. y = f(x) - p, p > 0 bang each tinh tien xudng dudi p don vi. y = f(x + q), q > 0 bang each tinh tien qua trai q don vj. y = f(x - q) , q > 0 bang each tinh tien qua phai q don vi. - Bai toan ve bien luan sd nghiem phuong trinh dang g(x.m) = 0 Dua phuong trinh ve dang f(x) = h(m) trong do ve trai la ham so dang xet, da ve dd thi (C): y = f(x). Sd nghiem la sd giao diem cua dd thi (C) vdi duong thang y = h(m). Dua vao dd thi va tuong giao vdi dudng thang thi cd sd nghiem tuong img can tim. 148 -BDHSG DSGT12/1- Download Ebook Tai: https://downloadsachmienphi.com Tron Bo SGK: https://bookgiaokhoa.com