T h s T o a n hoc - K s T i n hoc L E H O N G DUG - C h i i bieii LE null T R i PIIlTO]\ P H A P D A C B I E T G I A I TRU]\ HOC P I I O T I l ] \ Str D U N G PHU^dNG P H A P D I E U K I E N CAN VA D U GIAI TOAN NHA XUAT BAN HA NOI G l l T H I E U CHUNG Kill trciii tigng gic'ri lliieu tai ban doc bp tai lieu: i*iiiJOi\ i>iiAi» »Ac BI£:T Gi\i roAi\ I UUI\ H O C IMIO Tll6l\ Tliac sTTodn Jipc Le Hong Di'tc cliii bien D p t a i l i e u g o m I J tap: Cuo'n 1: Sir dung phuong phap luong giac hoa giai Toan Cuon 2: Sii dung phuong phap vecto giai Toan Cuon 3: Su dung cac phep bien hinh giai Toan Cuon 4: Su dung phuong phap toa giai Toan Cuon 5: Sir dung phuong trinh tham so Duong thiing, Duong tron, Eh'p va Hypebol giai Toiin Cuon 6: Sir dung phuong phap dat in phu giai Toan Cuon 7: Su dung phuong phap di6u kien can va du giai Toan Cuon 8: Su dung phuong phap ham so va thi giai Tosin Cuon 9: Sir dung gidfi han giai Toan Cuon 10: Su dung dao ham giai Toan Cuon 11: Sir dung may ti'nh giai Toan Muc tieu cua bp tai lieu tham khdo la cung cap cho cue Thdy, Co gido nipt bp bdi gidng cintyen sdu cd chat hd/ng va cho cac em hpc sinli Trung hpc phu thong yeu thicli mdn Todn mot bp tai lieu hoc tap bo ich Bp tai lieu ditac viet tren mot tit tudng liodn todn mdi me, c6 tinh sit pham, CO tinh long h(/p cao, tan dung ditcrc day di'i the manh ciia cac phuong phdp ddc biet de giai Todn Bp tai lieu chac cliaii phu hpp vdi nhieu ddi titpiig ban dpc tit cdc Thdy, Co gido den cdc em Hpc sinh lap 10, 11, 12 va cdc em chud'n bi dit thi mdn Todn Tot nghiep PTTH hodc vuo cdc Trudng Dai hpc Cuon Sir uiJNG iMiifoi\ i M i A i ' niv.ii KIEN CAIM V A nt oi.ii ro/iiv r//;'c/ ///a//// chi'i de: Chii de I: Sir diiiig phiraiig phap dieu kien can va dii giJi bai loan ve tinh chat nhat nghiem Chii del: Siidung phuang phap dieu kien can va du gitii bai loan ve tinh chat nghiem Chi'i de3: Sir dung phuang phtip dieu kien can va du giiii bai toiin ve tinh chat tham so inieii id (III lic't plncang phap gidi cho dang loan tan dung dupe day du ihe inanli cita phii'o'ng phap dlcit kien can vd dt'i Tdi Cling xin bay id tai day long biet on sdu sac td'l sif gli'ip do' dong vien llnli than cita lial nginyi Thdy ind Idl rat ini/c kinli Irpng, gdin GS.TS Trail Maiili Titan ngtiyen Fhd Gidiii doc Trung Tdiii KHTN & CNQG, Nhd gido I fit lit Ddo Thieii Klidi iigityen Hleu trudng TrUdng FTTH Hd Ndl Ainslerdaiii Citdl ciing, cito dii dd rat cd gang, ninfng thai khd Irdnh klidi iihuiig thlcit sol bdi nhifng hleu blel vd kinli nghlein liaii cite, id't inong nlidn dupe iiltilng y kien dong gop c/tiy bdu cita ban dpc gdn xa Mpl y kien dong gap xIn lien he tiri: Dia chi: Nhom lac gia Cu Mon - Nha sach Toan TMPT Cir Mon So 20 - Ngo 86 - Duong To Ngoc Van - Quan Tay Ho - Ha Noi Dien thoai: (04) 7196671 E-mail: cumon@hn.vnn.vn Ud iipl, ngdy llidng iidiit 2004 LE HONG DirC GlOl THifiU CHUNG MUC LUC CHIJ DE SU DUNG PHl/ONCi PHAP DIEU KIEN CAN VA DU (JIAI BAI TOAN VE TINH CHAT DUY NHAT NGHIEM Bai toan Bai toan Giai bai toan nha't nghiem cho phuang trlnh, bat phuang tiinh Giai bai toan nhat nghiem cho he phuang trlnh, he bat phuang trlnh 10 33 CHUD^2 SU DUNG PHUONG PHAP DIEU KIEN CAN VA DU (ilAI BAI TOAN VE TINH CHAT N(,HIEM Bai toan Bai Bai Bai Bai toan toan toan toan Giai bai toan ve tinh chat cac nghiem cho phuang trlnh Giai bai toiin ve tap nghiem Giai bai toan ve phuang trlnh qua Giai bai toan vi hai phuang trlnh tirang duang Su dung thi 66 77 92 95 126 CIIU Y)t (ilAI BAI TOAN PHAP VE TINH CHAT SO DU Sir DUNG PHUONG DIEU KIENTHAM CAN VA Bai toan Bai toan Phuang trlnh nghiem diing vai gia tri xiic dinh ciia tham so He nghiem diing vol gia tri xac dinh cua tham so TAI LIEU THAM KHAO 102 105 110 CHU D E I Str DUNG PHU'dNG P H A P DIEU KIEN CAN VA DU GIAI BAI TOAN V E TINH DUY NHAT NGHIEM M DAU Trong chu de se minh hoa each sir dung phuong phap dieu kien can va dii giai bai toan nhat nghiem cho phuong trinh, bat phuong trinh, he phuong trinh va he bat phuong trinh dugc chia hai dung: Dgiigl: Giai bai toan nhat nghiem cho phuong trinh, "bat phuong trinh chua tham so Dang 2: Giai bai toan nhat nghiem cho he phuong trinh, he bat phuong trinh chua tham so = — = — = BAI TOAN [| GiAi BAI T O A N D U Y N H A T N G H I E M C H O PHL/ONG TRINH, B A T PHUaNG TRINH • I V l (111 I : • CO n g h i e m d u y nhat mx^-2(m-l)xf, V a i yeu cau: fix, m) >() (liodcf(x, CO iii>hiem hat o BiioT 2: (1) c ^ m - = » m = Dieu kien du: V o l m = 1, ta c6: x"* = » (1), ta d i khang d i n h k h i x = cpCx,,) cung la n g h i e m cua (1) D o do, de he c6 n g h i e m d u y nhii't can c6: (2) T h a y (2) vao (1) ta xac d i n h dugc dieu k i e n can cho t h a m so m de (1) c6 n g h i e m nhat, gia sur meD,„ Dieu kien dir V o i meD,„, ta d i k i e m tra l a i tinh d u y nhat Chii y: Y e u ciiu tren hoan toan c6 the duoc thuc hien bang phuong phap dat an phu, cu the: Dat t = x% v o i t > 0, phuong t r l n h c6 dang: Buoc 4: phai xet cac phirong la k h o n g c6 tham so nhieu) K e t qua ciia D,„ cac gia t r i k h o n g Ket hop ba buoc giai tren ta t u n duoc dap so Tru'd'ni^ hop I V o i m = (2) » t - l = « t = ^-« ^ ' " ^ ^ Phuong t r i n h c6 hai n g h i e m phan biet Trifdiii^ hop V o i m * Phuong t r i n h (1) c6 n g h i e m d u y nhat II V i DU M I N H H O A 2{m i) T r u d c tien c h u n g ta m i n h hoa cac v l du sir d u n g t i n h chat h a m chSn de xac d i n h dieu k i e n can, tuc la xuat phat tii' nhan xet: • G i a su phuong t r i n h c6 n g h i e m x,, khang d i n h r i n g no c u n g se nhiin - x,, l a m n g h i e m • V a y de' phuong t r i n h c6 n g h i e m d u y nhat dieu k i e n la: - x„ = x„ » (2) f(t) = m t ^ - ( m - l ) t + m - = n g h i e m cho (1) T h o n g thuong buoc nay, ta c h i t r l n h , bat phuong t r i n h cu the (thuong hoac neu c6 t h l da duoc d o n gian d i budrc cho phep ta loai d i k h o i tap thich hop ciia m X = la n g h i e m nhat cua phuong t r i n h V a y , v i m = phuong t r i n h c6 n g h i e m d u y nhat x„ = (p(x„) =^ G i a t r i ciia x,, Bu(>c3: D o c h i n h la dieu k i e n can de phuong t r i n h c6 n g h i e m d u y nhat Dua tren t i n h chat d o i x u n g ciia cac bieu thiic glai t i c h c x„ = K h i do: Dieu kien can: Gia sir (1) c6 n g h i e m la x = x,„ k h i do: b m ( - X,,)' - ( m - 1)( - X,,)' + m - = - x„ = x„ nlid't" Dat dieu k i e n de cac bieu thirc (1) c6 nghla a =0 V a y de phuong t r i n h c6 n g h i e m d u y nhat dieu k i e n la: ta lliirc hien theo cac budfc sau: 1: + m - tuc la - x„ cQng la n g h i e m cua phuang t r i n h (1) m) ( m - ) ' (1) Gicii Dieu kien can: Gia sir (1) c6 nghiem la x = x„ suy - x„ cung la nghiem cua (1) Vay (1) CO nghiem nhat x„ = - x„ x„ = Khi do: • Thay x„ = vao (1), ta duoc: log , > (m - )^ « (m - 1)- < m = 1 + = m « m = Do chinh la dieu kien can de phuong tiinh c6 nghiem nhat Dieu kien dir V6i m = 3, phuong tiinh c6 dang: Do chinh la dieu kien can de phuong tiinh c6 nghiem nhat Dic'u kien dir Voi m = 1, (1) c6 dang: log2(2-Vx- + l) >C » VI: 4r- (1) ( ! ) « V - m m = Khi do: Vi du 2: ^ - Jx^ + > o VX- + 1» Ixl < o - x,, l a m n g h i e m Do p h u o n g t r i n h c6 n g h i e m nhat t h i di6u k i e n can la kien du: G i a sir m = 0, k h i (1) c6 dang: ^ kien can: N h a n xet rang neu p h u o n g t r i n h c6 n g h i e m x,„ t h i c u n g nhan x„= X = la n g h i e m nhat ( D o Dieu =cos2x CO n g h i e m d u y nhat thuoc ( - - , -) CLia giiii t i c h t r o n g p h u o n g t r i n h , bat phuong t r i n h c h i i n g ta da thitc h i e n duoc yeu cau " Tim trinh c6 nghiem dieu kien ciia tliam so de phifcfng trinh, bat phmng nhat" Cac V I d u tiep theo vSn v o i yeu cau nliir trcn x o n g de t i m dieu k i e n can c h u n g ta su d u n g cac phep bien d o i dai so de t h o n g qua n g h i e m x„ kien dii: V d i m = , k h i (1) c6 dang: = C0S2X (2) lam xua't h i e n n g h i e m (p(x„) V i d i i 8: Ta CO nhan xet: = V2-C0SX T i m m de p h u o n g t r i n h : x"* + m x ' + 2inx^ + m x + = > (1) CO n g h i e m d u y nhat VP = c o s x < l Gidi D o do: fVT = l ^^^^ y: N h u vay, t h o n g qua viec danh gia t i n h ch5n ciia cac bieu thirc Chii thu6c(-^,^) VT V a y , p h u o n g t r i n h c6 n g h i e m d u y nhat k h i va c h i k h i m = D o chfnh la d i e u k i e n can de phuong t r i n h (1) c6 n g h i e m d u y nhat yJl-COSK < • 2(cosx-l) = ci>x„ = V m - c o s O = cosO V m - l = < » m = Dieu lo)>xl VP = ( c o , s x - l ) V a y ( I ) CO n g h i e m d u y nhat k h i x„ = - x „ = x^ ] ^ m - c o s ( - x „ ) = cos2{ - x„) - x,| c u n g la n g h i e m Vi: kien van: G i a su (1) c6 n g h i e m la x = x,„ tiic la: ^ -2m + = o i n = x^ - 2co.sx + = x ' = 2(cosx - 1) VT V ' " - c o s x , | = cos2x„ = kien dii: V o i m = 1, k h i phuong t r i n h c6 dang: (1) Gicii Dieu « x „ D o c h i n h la dieu k i e n can de p h u o n g t r i n h c6 n g h i e m d u y nhat T i m m de p h u o n g t r i n h : Vm-cosx X|, K h i do: V a y , v o i m = p h u o n g i i l n h cc n g h i e m nhat V i d i i 6: - VP l ^ ^ U IVP = N h a n xet r i n g x = k h o n g phai la n g h i e m ciia phuong t r i n h cosx = l [2cos2x-I = l "^'"f'f' «cosx=l la n g h i e m d u y nhat ciia p h u o n g t r i n h « x = Dieu kien can: G i a i s i i (1) c6 n g h i e m x^^^O, suy x^ + m x,^ + m xfi + mx,, + 1=0 V a y , v o i m = p h u o n g t r i n h c6 n g h i e m d u y nhat 1s + m — +2m~ X„ + -L +m~ K h i phuang t r i n h c6 dang: =o r(t) = t ' + m t + m - = ^•i v2 (2) Phuang t r i n h (1) c6 n g h i e m d u y nhat » + m + 2m + m — +1=0 phuang t r i n h (2) c6 d i i n g gt n g h i e m thoa m a n Itl > - De nglu ban doe ticlani N h u vay, de t i m dieu k i e n cua tham so cho p h u a n g t r i n h h o i tiic la — c u n g la n g h i e m cua phuofng t r i n h y: ax'* + b x ' + cx" + bx + a = 0, v a i a V a y de phucfng t r i n h c6 n g h i e m d u y nhat dieu k i e n la: — • = X „ « X o + m + m + m + = m = - ^ Biioc I: N h a n xet l i i n g x = k h o n g phai la n g h i e m cua phuang trinh Bia'fc 2: Dieii kien vein: G i a i su (1) C O n g h i e m x,,, suy — (!) - m + m - m + = 0, v6 n g h i e m m = - ^ — nhat - - , ta c6: (l)x'- - x ' - x - - - x + 2 I 1= 02x'-x'-2x x + = ( X - )^(2x^ + 3x + 2) = X = la n g h i e m d u y nhat cung la n g h i e m c i a phuang t r i n h V a y de phuang t r i n h c6 n g h i e m d u y nhat dieu k i e n la: la dieu k i e n can de phuong t r i n h c6 n g h i e m d u y Dieu kien dir V o i m = = x,| » X|| = ±1 => G i a t r i t h a m so D o chinh la dieu kien can de phuang t i i n l i c6 nghiem nhat BIIOC 3: du 9: Dieu kien dir Thuc hien viec t h u l a i T\m m de phuang t r i n h : Isinx - ml + Icosx - ml = v V a y , m = - ^ p h u a n g t r i n h c6 n g h i e m d u y nhat Chii (1) hien theo cac bu6c: ±l V o i x„ = - 1, ta dugc: Vay, n g h i e m d u y nhat, bang phuang phap dieu k i e n can va du dugc thuc V a i x„ = 1, ta du-gc: (1) o • = 5i (1) C O d u n g m o t n g h i e m thugc (0, - ) y: Y e u cau t i e n hoan loan c6 the dugc thuc h i e n bang phuong phap dat an p h u , c u the: N h a n xet r i n g x = k h o n g phai la n g h i e m ciia phuang t r i n h Chia ca hai ve cua p h u a n g t r i n h cho x V O , ta dugc: x^ + m x + m + m - + -i^ = X 2 X Suy x" + ~ Dieu kien ran: G i a su (1) c6 n g h i e m la x = x,, suy X* + m ( x + - ) + m = X Gicii = t^ - V a y (1) C O n g h i e m d u y nhat k h i 7t C„ » X „ = Tt 4• T H l / V! 17 T h a y x„ = ^ vao (1), la dugc: T h a y x„ = ^ vao (1), la duac: I s i n - - m i + l c o s - - ml = V2 « > l ^ 4 , V2 , Im - — V2 I = — 2 -ml + l ^ - mi = V2 m = DieII kien dir V o i m - 0, k h i (1) c6 dang: kien di't ^ Vai m = xe((),^) (1) Isinxl + Icosxl = •s/2 o V2 sin(x + - ) = V2 o X + - ^ - + 2k7t o c?" •» o =2 D o do: xe((),") (2) « > « > X = ^ la n g h i e m d u y nhat ^/tgx - Tcotgx = ! tgX = la n g h i e m d u y nhat cua phuang t r l n h V a y , v o i m = p h u a n g t r l n h c6 n g h i e m d u y nhat Isinx - N/2 I + Icosx - V2 I = N/I sinx + cosx - -Jl - gicii tmmg ti( Vi dii 11: T i m m de phuong t r l n h sau c6 n g h i e m d u y nhat; nhinren khocing (0, ^ ) ^ + ^42^ + 4^ + ^Jl^K = in (1) Gicii DieII kien can: G i a su phuang t r l n h (1) c6 n g h i e m la x = x,, suy - x,| c u n g la n g h i e m cua (1) T i m m de p h u a n g t r l n h c6 n g h i e m d u y nhat thuoc (0, ^ ) : ^tgx-m + ^cot g x - i i i =2 (1) Gicii V a y (1) C O n g h i e m d u y nhat k h i x„ = - x„ » x„ = T h a y x„ = I vao { ! ) ta dugc m - Dieu kien can: G i a sir (1) c6 n g h i e m la x = x,,, tiic la: V'gX() - m + ^cot gX() - m y - X|| cung la n g h i e m cua (1) V a y (1) C O n g h i e m d u y nhat k h i X|| _ — Tt — — X,, 18 (2) , X = - + 2k7t V a y , v d i m = hoac m = >/2 p h u o n g t r l n h c6 d i i n g n g h i e m thuoc V i d u 10: = V T = -y/igx + ^cot gx >2^^/tgx.ycoTgx^ sin(x + - ) = V o i m = V2 (1) « + Vcotgx A p d u n g bat dang thiic Cosi, ta dugc: sinx + cosx = V2 xe((),?) • m = thuoc k h o i i n g (0, ^ ) k h o i i n g (0, ^ ) • - "11 = V l - m = « D o c h i n h la d i e u k i e n can de phuong t r l n h (1) c6 n g h i e m d u y nha^t m = V2 D o c h i n h \i\u Icien can de phuong t i l n h c6 d i i n g n g h i e m thuoc Dicii tg ^ - m + ^cot g _ 7t X|| — — • D o c h i n h la d i e u k i e n can de phuang t n n h c6 n g h i e m d u y nhat Dieu kien di'i = V o i m = 4, k h i (1) c6 dang: ill + ifl^ = + V7 + (2) A p d u n g bat dang thCrc B u n h i a c o p x k i , ta dugc: + V ^ < va ^ + ifl^ < n Vi du 13: Do d o : T i m m de p h u o n g t r i n h sau c6 n g h i e m d u y nhat: mx(2 - X) = ix - II (1) Gicii Oicu kien can: X = la n g h i e m d u y nhat cua phuong t r l n h mx„(2 - x„) = lx„ - 11 « V i i y , v o i m = phuang t r i n h c6 n g h i e m d u y nhat V i d u 12: V a y de phuang t r i n h c6 n g h i e m d u y nhat dieu k i e n la |X|, K h i do: (1) « G i a sii (1) c6 n g h i e m la x = x,, suy la - al + IX|, - x„ = x„ o (1) Gicii Dieu kien can: m = Dieu kien di'i: V o i m = 0, ta c6: (1) Ix - 11 - « => a + b - X|| cung la n g h i e m cua (1) X = la nghiem d u y nhat cua phuang t r i n h V a y , v d i m = phuang trinh c6 n g h i e m d u y nhat V i du 14: V a y (1) CO n g h i e m d u y nha't k h i T i m m de phuang t r i n h sau c6 n g h i e m d u y nhat: , a+b = a + b - x„ x„ = • Thay x„ = x,| = D o c h i n h la dieu k i e n can de phuang t r i n h c6 nghiem d u y nhat - bl = c l(a + b - x„) - al + l(a + b - x„) - bl = c X|, m [ - (2 - x„)](2 - x„) = 1(2 - x„) - 11 tiic la - x„ cung se la n g h i e m cua phuang t r i n h T u n a, b, c de phuang t r i n h sau c6 n g h i e m d u y nhat: l x - a l + l x - b l = c G i a i sii (1) c6 n g h i e m x,„ ta c6: 3IX-21 = m - Gicii vao (1), ta dugc: Dieu kien cc'in: Gia su phuang t r i n h c6 n g h i e m la x = x,, suy c = la - bl D o c h i n h la dieu k i e n can de phuang t r i n h c6 n g h i e m d u y nhat => - X|, cung la n g h i e m cua (1) Diet! kien di'i V a y phuang t r i n h c6 n g h i e m d u y nhat k h i G i a sir c = la - bl, k h i (1) c6 dang: Ix - al + Ix - bl = la - bl « (X • x„ = - x„ « Ix - al + Ix - bl = l(x - a) - (x - b)l - a)(x - b) < (2) N e u a ;^ b (ta gia su k h i a < b), k h i do: T h a y x„ = vao p l i u a n g t r i n h , ta duac m = D o c h i n h la d i e u k i e n can de phuang t r i n h c6 n g h i e m d u y nhat Dieu kien du: G i a su m = 1, k h i phuang t r i n h c6 dang: (2) (x-a)- 3'" = 1» Ix - 21 = X = la nghiem nlia't V a y , v o i m = phuang t r i n h c6 n g h i e m d u y nhat Vi du 15: T i m m de phuong t r i n h sau c6 n g h i e m d u y nhat: 2mx,2-x, ^ l x - l l + ,ii_ Giiii Dieu kien cc'iu: G i a i su phuong t r i n h c6 n g h i e m x,,, ta c6: 2nix„(2-x„) _ ^Ix,,-!! + j-j^ < ^ 2"i(2-x„)|2-(2-X|,)| _ tiic la - X|, cLiiig se la n g h i e m cua phuong t r i n h 3l(2-Xo)-ll + • Vay de (5) nghiem diing vai moi te [0, 3] Bien doi ve phai ve dang: VP = - 2x + m = (X - +m- 1>3 m > y(3) m > Suy ra: Vay, v6i m > thoa man dieu kien dau bai Vi clu 6: V(2 + x ) ( - x ) < x^ - 2x + m Tun m de phuong trinh sau nghiem diing Vx > 0: Vay, vdi m>4 bat phircmg tiinh nghiem diing V x e [ - 2, 4J Vx' + x - n r + 2m + = X + m - (1) Chii y: Bai toan tren c6 the dugc thirc hien biing cac each sau: Cchli J: Su dung phuong phap bien doi tuang duong Bien doi bat phuong trinh ve dang: x 2x + m>0 (2) [Dieu kien cdir Gia sii (1) c6 nghiem Vx>0 =i> x = la nghiem Ciia (1), "khi do: (1) V- (I) m- + 2m + = m - (2 + x ) ( - x ) < ( x ^ - x + m)^ (3) Vay (1) nghiem diing V x e [ - 2, 4] < o -m~ + 2m + = ( m - r m = (I) nghiem diing V x e [ - 2, 4] (2) nghiem diing Vx e [-2,4] m - >0 Do chinh la dieu kien can de phuong triuli nghiem diing vdi Vx>0 )/('// kien dir V d i m = 3, (1) c6 dang: m > X>(1 , (3) nghiem dung Vx e [-2,4] V x ' + x + l = x + l x + l = x + l < = > = luon diing Vay, voi m>4 thoa man dieu kien dau bai Vay, voi m = phuong trinh nghiem diing Vx>0 Ccic/i 2: Su dung phuong phap dat an phu ciing vdri tarn thiJc bac hai f CVi/i y: V o i bai toan c6 nhieu hon mot tham so ta se thay tam quan Dat t = V(2 + x ) ( - x ) , voi x e [ - 2, 4] ta nhan dugc dieu kien ciia t la 0 - t (i) 2) chiia doan [ - , I ] " Gicii Dieu kien can: Bat phuang t r i n h n g h i e m d i i n g v a i V x [ l , 3] => n g h i e m d i i n g v a i X = 1, x = 2, tiic la ta c : I2in+17l t = ^ va t D o chfnli la dieu kien can de nghiem ciia bat phuang t i i n l i chiia [ - - < x ' - 8x + < (x-2r G i a su (1) c6 n g h i e m V t e [ ^ , > — kien du: V a i m = - 8, ta c6: (1) « kien can: - 8 Vay, voi m< - nghiem ciia bat phuang trinh chua [ - , 11 2x^-8x + < Chii y: Cung c6 the khong can su dung phuong phap dieu icien can va dii Irong bai toan tren, cu the: - < 2x' - 8x + < ix-2)^ >0 < x < x^ - x + < Vay voi m - - bat phuong trinh nghiem diing V x e [ l , 3] yi du 14: T i m dieu kien ciia m de bat phuung tiinh: Bien ddi bat phuang trinh ve dang: Ig V(2 + x ) ( - x ) ( i -t)^c:>f(t) = ( m - l ) t + < (*) Vay de nghiem cua bat phuong trinh chiia [ - , 1] dieu kien la: (1) X', ghiem diing voi moi x e ( - 2, 4) idi ^ Bien doi bat phuong trinh tuong duong voi: f(l) Do la dieu kien can de bat phuong trinh nghiem diing voi V x e ( - 2, 4) Dieu kien ccin: Gia su (1) c6 nghiem Vx > do: x = la nghiem ciia (1), Dieu kien du: Gia sir m>4, do: • ( l ) » m + = m = A p dung bat ding thiic Cosi cho ve trai, ta duoc: x/T [T-, ; 0 Dieu kien du: Vdfi m = 0, (1) c6 dang: • x>() = x o Bien doi ve phai ve dang: VP = x ' - 2x + m = (X - ) ' + m - > x = x l u n diing • Vay, voi m = phuong trinh nghiem diing voi Vx>0 Vi du 13: _ ^ = Suy ra: V(2 + x ) ( - x ) < x^ - 2x + m T i m m de bat phuong trinh sau nghiem diing V x [ 1, 3]: i x ^ m ( x i H i i < _ ( m + ) ( x ^ - x + 2) •::) (1) Vay, voi m>4 bat phuong trinh nghiem diing vdi V x e ( - 2, 4) Chu y: Co the sii dung gtln, gtnn ciia ham s6' de giai v i du tren, cu the: Gicii Dieu kien can: Bat phuong trinh nghiem diing V x e [ l , 3] => nghiem diing voi X = x = 2, tiic la ta c6: I2m+17l0 i d u 3: = -4 vc// moi xeD " ) C h u n g ta d i x e m Cho p h u o n g t r i n h va bat p h u o n g t r i n h : > < in < Vx-l+2nWx-2 (1) + Vx-i-2nWx-2 = (2) Ix^ + 3x + 2l3x = k7rox=-!^, R G Z Bai tap 2: T i m gia tri ciia m de hai phuang trlnh tuang duang 2cosx.cos2x = + cos2x + cos3x Do ( I ) & (2) tuang duang trudc het can x = (la mot nghiem cua ho X = ~ ) cung la nghiem ciia (2), tire la: 4cos^x - cos3x = acosx + (4 - a)( + cos2x) Bai tap 3: T i m gia tri ciia m de hai phuang tnnh tuang duang m.cosO + Iml.cosO + cosO = m + Iml = m < cos3x = 4cos(37t + x) Do chinh la dieu kien can ciia a va b mcos^x + (1 - m ) s i n ( ^ + x) = Dicu kien dit • Bai tap 4: V a i m = 0, ta dugc: (2) ! - 2sin^3x = sin3x = » x =k7r«x= sin3x = ainx + (4 - 2a)sin^x ~.k&Z Bai tap 5: Xac dinh a, b de hai phuang trlnh sau tuang duang: asin2x -2-^3 = a A/3 Vay m = thoa man dieu kien dau bai • ' tuong duang vai bat phuang trlnh Vx^ +4x + + x + 1>0 - 2.sin^3x + 2msin3x.sinx = (sin3x - msinx)sin3x = Bai tap 7: (De 10): T i m m de hai phuang trlnh sau tuang duang: sin 3x = sin X - sin 2cos2x.cosx = + cos2x + cos3x x- ni sin x = 4cos^x - cos3x = mcosx + (4 - m)( + cos2x) Bsii tap 8: (De 40): Tun m de hai phuang trlnh sau tuang duang: X = 3x - kn - 4sinx Bai tap 6: T i m m de phuang trlnh: lx-ml-lx+11 = m.co.s2x - m.co.s4x + cos6x = i cos6x - - m(co.s4x - cos2x) = o COSX sin2x + cos2x + b V2 + = 42 sinx + 2cosx(co'sx + b) Vdri m0 x-l>0 3-Vx-l « l < x < >0 Bien doi (2) ve dang: V6-X = 3- V x ^ < » V6-X + >/x^ =3 ( - x ) + ( x - l) + V ( - x ) ( x - l ) = o II v i D U M I N H H O A V(6-x)(x-l) X = Vi (111 1: o T i m x de phuong trinh sau nghiem diing vdi moi a: log^,2^2^N/x + - l ) = l0g^2^2^2(2-V5^) Gidi (1) b (6 - x)(x - 1) = « x' - 7x + 10 = o = 2 x =5 Su dung dieu kien can va dii de thuc hien Dieu kien can: Phuang trinh nghiem dung vo'i Vm > 0, truac het nghiem Dieu kien can: Gia sir (1) nghiem dung voi mgi a => diing voi a = V a i a = 0, ta duoc: dung vai m = "x = ( l ) o l o g ( V ^ - l ) = log2(2- ^/^) 0, bai vcfi in = mau thuin 10? 103 • V o i X = 5, phucfiig t r i n h (1) c dang: BAI T O A N l o g j l = log2^„l (luon diing) HE N G H I E M D U N G V O l V a y , X = n g h i e m d i i n g phucnig t r i n h da cho v d i m o i m > III G l A TRI XAC D I N H CUA T H A M SO BAITAPDfiNCJHI I F H i r O N C ; P H A P Biii tap 1: Cho p h u o n g t r i n h 2log^^2_^2^4-V7 + 2x) = log^^2^2^2^4-3x) a G i i i i phuong t r i n h v d i a = b T i m cac gia tii cua x ngliiem diing phuong tiinli da cho voi moi m > Bai tap 2: Cho p h u o n g t r i n h l o g ( a V + 5a^x + V - x ) = log O - V x - l ) a Cho he chiia hai t h a m so a, b, v d i yeu cau: " Tim a de lie c6 nghiem vai moi h lliiigc D,, " ta ihuc hien theo cac budc: Bum-1: Dat d i u k i e n de cac bieu thuc ciia he CO ngliTa Bum-2: Dieii kien can: G i a sir he n g h i e m d i i n g v d i V b e D ^ , suy no n g h i e m d i i n g v d i b„eD,„ G i a i p h u o n g t r i n h v d i a = • • b T u n cac gia t i i ciia x ngliiem diing phuong tiiiih da cho vol m o i m > Q Biioc 3: G i a i he v d i b = b„ => g i a t r i ciia a„ Dicht kien dir Thuc hien phep k i e m tra v d i a = a„ II V i D U M I N H H O A Vidul: T i m a de he phuong t r i n h sau CO n g h i e m v d i m o i b: (x^ + i r ' + ( b ' + l ) y = a + bxy + x"y = Giai Dicht kien can: D o he c d n g h i e m v d i V b , nen phai c d n g h i e m k h i b = K h i d d he CO dang: (x- + ir' = a + x^y = fa = O JL^ =0 a + x"y = : a=0 a = 1' V a y dieu k i e n can la a = hoac a = Dieu • kien can ^ V d i a = he c d dang: (b2 + i)y = i (1) bxy + x'^y = (2) He tren k h o n g the c d n g h i e m v d i m o i b, bdi k h i h^O (i)^y = o N g h i e m k h o n g thoa m a n (2), d o vay he v n g h i e m 105 • N h i l n xet rang he tren c h i c6 n g b i e m k h i I - 2b > b < ^ Vdfi a = he c dang: V a y a = k h o n g thoa m a n • V d i a = - , k h i he c6 dang: bxy + x ' y = he l u o n c6 n g h i e m x = y = v d i m o i b V a y , v o i a = he c6 n g h i e m voi m o i b V i d i i 2: Xac d i n h cac gia t r i ciia a cho he sau day c6 n g h i e m v o i moi b ^ - o g X + log2y = I N h a n xet rang he tren l u o n nhan x = & y = l a m n g h i e m V a y v o i a = - he c n g h i e m v d i m o i b Vx^-2b^-l-v'(a-l)by =x-l (I) ax + by - = Vi dii 4: V a i ' a > 0, cho he p h u o n g t r i n h : x + y = b^-b+ l Gicii DieII J Vx" »- -1 = (I) « ax - = T i m a de he c6 n g h i e m v a i m o i b e [ , l ] ] ^ X =l [ax - = [a = Gidi B i e n d o i t u a n g d u a n g he ve dang: V a y a = la dieu k i e n can de he c6 n g h i e m v o i m o i b Dieu a-^+ay=2 kien can: He c6 n g h i e m v o i m o i b => c6 n g h i e m v o i b = 0, k h i do: b^-b+l kien di'i: V d i a = , he ( I ) c6 dang: fx-i>0 Vx^-2b^-l = X -1 X + by - = X = b^ + I « \2 X « + by - = aUa>'=r X = b^ + X + by - = i fx - b " + It nha't mot n g h i e m la • ^ ~ [b-'+by = ' ' K h i a\* la n g h i e m d u a n g ciia p h u a n g t r i n h : t ^ - l t +a^'-^^' = kien can: H e c6 n g h i e m \6\, 1] • Dii'u p h u a n g t r i n h ( ) c6 hai n g h i e m d u a n g V b e [ , 1] iy = - b V a y , he p h u o n g t r i n h c6 n g h i e m v i m o i b k h i a = Vi d u 3: a\a-^=a*^'-''^' => p h u a n g t r i n h ( ) c6 hai n g h i e m d u a n g v d i b = va b T i m a de' he sau c6 n g h i e m v d i m o i b: 2'''"'^-^'+b(a + l)log.^y ^ a ^ • V a i b = , theo a) ta c6 d i e u k i e n l i i < a < • V a i b = ^ , d i e u k i e n la p h u a n g t r i n h (I) — 16 ( a - l ) l o g X + log5y = Gidi Dieu kien can: He c6 n g h i e m v a i V b => c6 n g h i e m v i b = 0, k h i do: (I)« 1^- l = a" => a = ± 4t ( a - l ) l o g X + log2y = D o c h i n h la d i e u k i e n ciin de he n g h i e m v a i V b Dieu • kien di'i iog2 y = 106 2hl"y3''+2b = l [iog^ y = =0 i 4a4 >0 A>0 V a i a = , k h i he c6 dang: 2''''*'%2blog^y = l + a"* hai n g h i e m d u a n g S>0 b'''"'^-^'= l - b I log2 y = P>0 o i > a>0 c^O0, V b e [ , 1] Do chinh la 6'i6u kien can de he nghiem voi Vbe [0, 1] Xet ham so y = b' - b + (1 + log J 6) Dieu kien dir Gia sir < a < —\j= , k h i xet (1) ta c6: 32V2 Voi Vbe [0,1] • Mien xac dinh D = [0, 1] • Dao ham: y' = b - 1, A he c6 nghiem Vay, voi < a < — — • c^y_ >0c;>^+log l6>0«a0 A>0 s > , V b e [ , 1] o f>0 P>0 Giiii he vdi a = b = b Tim a de he sau co nghiem vdi moi b Hai tap 2: , V b e [ , 1] Xac dinh cac gia tri ciia a cho he sau cd nghiem vdi moi b: la(x~+y") + x + y = b a'''-''-'>0 c:> a''"-''-' < — , V b e [ , 1] 16 He C O nghiem vd; V b e [0 ] C O nghiem vdi b = => (theo a) < a < — , la dieu kien can ciia a 108 a y-x =b (3) T A I I.II;:U TIIAm K i i A o Tnin Van Hao ( chu bien ) - Cluiyen de luyen thi vao Dai hoc Dai so - Niia xuat ban Giao due -2001 Nguydn Van Mau - Phuong piicip giai phuor.g innli va bat plurong trinhNha xuat ban Giao due - 2001 Phan Huy Khai - Phirong phap dieu kien cttn va du de bien luan he c6 tham so - Nha xuat ban Giao due - 2001 NguySn Dire DOng & Nguyen Van VTnh - 15 phuong phap chuyen de tarn tlu'rc bac va cac ling dung dae sflc - Nha xuat ban Tie - 2000 TiSn Phirong - Le Hong Diic Dai so so cap Nha xuat ban Ha No! - 2003 Phuorng phap dac biet giai toan T H P T sif Dmci i>iiLrOi\fii I M I A P » H U K i f \\iV v A » i j G i A i TOA^ Chiu tntch nhiem xuat ban : Gidm doc N C U Y f i N K H A C O A N H Bic'ii tup noi clung: L E BiCH NGOC Tiinli bay hia: L E SICH NGOC Che ban: L E HUtJ T R I L I CUOI Nhdm CiiMon luon san long gmi dap mgi tlidc mdc cua cac em hoc sinh Vii dpcgj'a ve noi dung cua cuon hii lieu Mgi clii tiet xiii lien lie true tiep tcfi: Ths L e Hong Due So nha 20 - Ngo 86 - Duong T o Ngoc Van - Tay Ho - H a Noi Dien tlioai: 04 7196671 ... Quan Tay Ho - Ha Noi Dien thoai: (04) 7196671 E-mail: cumon@hn.vnn.vn Ud iipl, ngdy llidng iidiit 2004 LE HONG DirC GlOl THifiU CHUNG MUC LUC CHIJ DE SU DUNG PHl/ONCi PHAP DIEU KIEN CAN VA DU (JIAI