Tài liệu Phương trình, bất phương trình hữu tỉ - vô tỉ - mũ logarit

512.9 PH561T •4 (CB) • ThS. PHAN VIET BAC MHAN - CN. LE PHUC LUf J f HAT l»HI]liSS •nrnCo ^ D U N G CHO HS GIOI THI TRirOfNG C H U Y E N ^ O N THI THPT QUOC GIA (2trong1) I^NrtA'XUAT BAN • C QUOC GIA HA NOI 5^1. g TS. LE XUAN S d N (CB) • ThS. PHAN V I E T BAC ThS. TRAN NHAN - C N . LE PHUC L Q HUUTI VO Tl MO LOGARIT ^ D U N G CHO HS GIOI, THI TRI/flfNG CHUYEN y ^ O N THI THPT QUOC GIA DTR: N H A X U A T B A N D A I H Q C Q U O C G I A H A NQI NHA XUAT BAN DAI HOC QUOC GIA HA NQI 16 Hang Chuoi - Hai Ba Triing - Ha Npi Dien thoai: Bien t$p - Che bkn: (04) 39714896; Quan ly xuat ban: (04) 39728806; long bien tSp: (04) 39715011 Fax: (04) 39729436 * * * Chiu trdch nhi^m xuat ban: Gidm doc - Tong bien tdp: T S . P H A M T H I T R A M NHA SACH HONG A N Trinh bay bia: NHA SACH HONG AN ban: Che tdp: Bi^n VAN ANH - PHUONG ANH ^' Doi tdc lien ket xuat ban: NHA SACH HONG A N SACH L I E N K E T PHaONG TRINH - BAT PHUONG TRINH HLJfU TJ, VO TJ, MU, LOGARIT Ma s5: 1L - 99OH2015 In 2.000 cudn, kh6 17 x 24cm tgi COng ti Cd phin VSn h6a VSn Lang. Dia chl: S6' 6 Nguy§n Trung Tri;c - P5 - Q. Binh Thanh - TP. Ho Chi Minh So xua't bSn: 351 - 2015/CXB/4 - 74/DHQGHN, ngay 09/02/2015. Quyg't djnh xuS't ban s6: 120LK-TN/QD - NXBOHQGHN. In xong va nOp liAi chieu quy II n3m 2015. Cac em hoc sinh than men! Phuomg trinh, bat phuang trinh la nhOng noi dung can ban trong chuong trinh toan ph6 thong. Co dugc ky nang t6t trong viec giai phuang trinh, bSt phuang trinh se khong nhung gop phan quan trong dS hinh thanh va phat trign nang lire giai quyet van de ciia hoc sinh ma con giup cac em dat k6t qua t6t trong nhtjng ky thi quan trong nhu: thi vao truang Chuyen, thi dai hoc, thi hoc sinhgioi cac d p . Vai muc dich ay, chung toi bien soan cuon sach nay nham cung cap cho ban doc mot he thdng bai tap phong phu, da dang vai nhi§u bai mai la va cac phuang phap giai hieu qua ve phuang trinh, bdt phuang trinh. Noi dung ciia cu6n sach dugc trinh bay thanh ba chuong: Chmmg 1 de cap den phuang trinh bat phuang trinh dang da thuc va huu ty; ChiroTig 2 de cap den phuang trinh, bat phuang trinh v6 ty; ChiroTig 3 va Chirong 4 theo thu tu de cap den phuang trinh, bat phuang trinh mu va logarit. Trong tung muc, tung phuang phap deu c6 cac vi du minh hoa tieu bieu; c6 phan bai tap de ban doc ren luyen; c6 phin huong din giai bai tap ngay sau do de ban doc tham khao, so sanh vai lai giai cua minh. Sau moi chuong deu c6 phan bai tap tong hgp, phan Ian la bai tap hay va kho. Chung toi hy vgng rang cuon sach "Phuang trinh, bat phiromg trinh va phuang phdp gidi" se thuc su huu ich cho cac em hoc sinh cung nhu cac thay, CO day Toan a truang pho thong. Du da hit sue c6 ging trong qua trinh bien soan, nhung bg sach kho tranh khoi nhung thieu sot nhat dinh. Cac tac gia chan thanh cam an y kien dong gop cua cac thay giao, c6 giao va cac em hoc sinh gan xa de Ian tai ban bg sach se dugc hoan thien hon. Mgi y kien dong gop cho tac gia xin quy ban dgc gai ve: [email protected] CAC TAC GIA 1 ChUctng 1. PHLfdNG TRINH, BAT PHtTdNG TRINH HlTU TI §1. TAM THlTC, PHirONG TRINH, BAT PHlTOfNG TRINH BAG HAI 1) DIU cua tarn thipc bac hai Tom tit ly thuyet 7.7. Dinh ly ve ddu cua tarn thuc Cho tarn thuc bac hai f{x)^ax^+bx + c,a^Q. Dat A = b^ - 4ac. Khi do: NSu A<0 thi af (x)>0 vai moi xeR; NSu A = 0 thi af (x)>0 voi moi x^2a Ngu A>0 thi af{x)>0 vai moi x G ( - O O ; X , ) U ( X 2 ; + C O ) va af(x)<0 vai moi xe{x^;x2), trongdo x, 0,VxeM<» a>0 A<0 \A| a>0 3./(x)>0,VxeM<:> a>0 A<0 A<0 1.3. Gid tri Ian nhdt, gid tri nhd nhdt cua tam thuc Cho tam thuc bac hai f(x) = ax'^+bx + c,a^ 0. Ta c6 2 b^^ fix) = a x + — + c, nen I 2aj V 4a b_ Vai a > 0, f{x) c6 gia tri nhoJ nhat la c , dat dugc khi x = - la 4a 5 Vdfi a < 0 , / ( x ) CO gia t r i Ion nhdt la V i d u 1 . Cho cac so thirc x,y,z c- 4a , dat dugc k h i X = - 2a thoa man x + y + z^\. T i m gia t r i lom nhdt c u a b i d u t h u c A = 9xy + I0yz + 1 I z x . LcigiaL Thay z = \-x-y vao ^ t a c 6 A = 9xy + \(iyz + nzx^9xy + z(\Qy + \\x)^9xy + (\-x-y){\Qy + \\x). K h a i trien va n i t gon ta c6 ^ = - l l x ' - 1 0 / + l l x + 10>;-12xv • D o d o \ +{\2y-n)x - ---^ =-llx'+(ll-12>;)x-I0/+10>;. + \0y^ -\0y + A = 0. De CO gia t r i Ion nhat ciia A t h i phuong trinh c6 nghi?m. T a c6 A > 0 < » - 2 9 6 / + 1 7 6 ; / + 121-44^>0. l l V 495 495 74 2 22 121 74 ' D o do ^ < y- y + — V + < 11 27 11 37 296 148 148 27 25 11 Dku dang thuc xay ra k h i x = — ; v = — : z = • ^ ^ 74 37 74 495 0V a y gia t r i I o n n h i t ciia A la 148 V I d u 2. Cho hai s6 thuc x,y thay d6i thoa man x^ +y^ = \. T i m gia t r i Ion nhat va nho nhat cua bieu thuc P = 2 • 1 + 2xy + 2y (Di thi dgi hoc khoi B 2008) L&igidL Voi = 0 ta CO = 1 nen P = 2 . X 2fx^+6xv) Vai>;^0,datr = - t a c 6 y Dodo P[t'+2t 2(x^+6xy) P^-^ - ^ =- ^ ^ = l + 2xy + 2 / x'+2xv + 3 / + 3) = 2t^+\2tc>{P-2y+2{P-6)t 2/^+12^ 7 • t^+2t + 3 + 3P = 0. , _ ^.l. V a i P = 2, phuong trinh c6 nghiem t = — . 4 Voi P^2, phuomg trinh c6 nghiem khi vachi khi A' = - 2 P ' - 6 P + 3 6 > 0 o - 6 < P < 3 . P = ?> )&h\ = —F=,y = ^ = hoac x = — = , j j ; = - VTo • 3 2 3 P = -6 khi x = — v = — p = hoac x = — i = , y VTI 713 • Vl3 = 2 — 1 = . Vn Ket hop lai ta c6 gia tri nho nhdt cua P la - 6 , gia tri Ion nhdt cua P la 3. Vi du 3. Cho a,b^O. Tim gia tri nho nhdt cua bi^u thuc 1 b P = a' +b' +^ + - . - . a a L&igidL Xem P nhu la mot tam thuc bac 2 doi voi bien b. Taco P = b' +2b— + -K + +a = b+ 2a la 4a' 4a' + •4a' 4a' D4U bang xay ra khi +a + a'>2. -.a' \4a- b= 2a a' = 4a' 2 Vi du 4. Cho cac so duong a, b, c thoa man a + b + c = 3. Chung minh rang 9 a + ab + 2abc< — . 2 . Laigiau Tugiathiet 6 = 3 - a - c.Tac6 9 9 a + ab + labc < — • o a + a ( 3 - a - c ) + 2ac(3 - a - c) < —. Dat / ( a ) = (2c + l)a^ +(2c^ - 5 c - 4 ) a + ^ > 0 . Tachung minh / ( a ) > 0 . Ta CO / ( a ) la mot tam thuc bac hai c6 he so ciia a' la 2c +1 > 0, va lai c6 A = ( 2 c ' - 5 c - 4 ) ' - 1 8 ( 2 c + l ) = ( 2 c - l ) ' ( c ' - 4 c - 2 ) < 0 do 0 < c < 3 . 3 1 Tir d o / ( a ) > 0. Dau bang xay ra khi a = — ;6 = l:c = —. 2 2 Vi du 5. Cho 4 s6 thuc a,6,c,£/thoa man: + 6^ = 1; c - 36;abc = 1, Chung minh rang— +6 +c >ab + bc + ca. L&igidL— Ta+ c6 b^+c^>ab + bc + ca<^(b + cY-a(b >0 + c) + - 8 <;=>(Z) + c) -2-{b + c) + — + >0 b+c + • 12a > 0, luon dung vi > 36. Vi du 7. Cho hai s6 x,y thoa man -2xy-2x + 4y-7 = 0. Tim gia tri cua x \ihi y dat gia tri Ion nhit. (Di thi tuyin sinh THPT Chuyen Qudng Ngai 2013) LcfigidL TsiCO x^+y^-2xy-2x +Ay-1 = Q ti. <::>x^ - 2{y + \)x + y^ +Ay-1 = 0 De ton tai gia tri cua x thi phuomg trinh tren phai c6 nghiem, do do A' = (>; + 1)^ - - 4;; + 7 > 0 O 2>; < 8 >; < 4 . Khi y = A thay vao phuong trinh ta c6 x = 5. Vay X = 5 thi dat gia tri Ion nhdt. Bai tap phan 1.1 +z =\ loa man 1. Cho x,y,z thoa man < [2x'+3/+4z'=3 Tim gia tri Ion nhdt, gia tri nho nhat cua y. 2. Cho 3 so x,y,z thoa man 1'xxy++y_yz+ z+=zx5= 8 .Chungminhrang \;^ + 4 (1 - X- =3 yf < = > 6 x ^ + 7 / - 8 x - 8 > ; + 8xv + l = 0 ,. ^ . ^ 4> <=>6x^+8(>;-l)x + 7 / - 8 > ; + l = 0 . , i^-xviv D I phuong trinh c6 nghiem ta c6 A ' = 1 6 ( : i ; - 1 ) ' - 6 ( 7 / - 8 3 ; +1) = - 2 6 / + 1 6 > ; +10 > 0 < >^ < 1. Voi y = l khi do ta CO x = z = 0 . Vay gia tri lom nhdt cua 3^ la 1. ,, Voi y = —— khi do x = — , z = — . Vay gia tri nho nhat cua >> la - ^ 13 13 13 13 2.Tir he ta c6 Do do [jj;z = x ^ - 5 x + 8 ' >'z = 8 - x ( > ; + z ) J>' + z = 5 - x j +z =5- x xjr-: ,«f c or - (5 - x ) / + x^ - 5x + 8 = 0. z la hai nghiem cua phuong trinh Phuong trinh c6 nghiem A' > 0 <=> (5 - x)^ - 4(x^ - 5x + 8) > 0 " ' ^ <::>-3x^+10x-7>0<»l; = 0 , t a c 6 x ; t O , P = l . 'T' ' ' ^^r^^^i ^il^ ml 10 r ^^ X Vai ;;7tO,tac6 P = / \ y r-/ +l y ~ = ^ X , + - +1 -,vai X t= -. ^ +^ + 1 J' y Taco y.rs t — f +1 ^=^r^y^<^/'(r'+/ + i) = ^'-^ + i < » ( p - i ) ? ' + ( p + i ) r + ( p - i ) = o. De ton tai gia t r i lom nhSt, gia t r i nho nho n h i t cua P, phuang t r i n h n o i tren phai CO nghiem, k h i do A'= (P + 1 ) ' - 4(P - 1 ) ' > 0 V a i P = ^ xay ra k h i r = 1 hay -3P^ + lOP - 3 > 0 <=> ^ < P < 3 . = X 0. ^ 'v; u V a i P = 3 xay ra k h i ^ = - 1 hay x = ->» ^ 0 . Vay, gia tri Ian nhSt cua P la 3, gia tri nho nhdt cua P la ^ . 4. T a c o \ a ' < 3 a - 2 . b,c,d. P-a^ +6^+c^ + <3(a + 6 + c + ^)-8 = 10. V a i a = 2,6 = 2,c = 1,^/ = 1 hoac cac hoan v i t h i P = 10. Vay gia tri Ian nhdt ciia P la 10. ,2 5. Tir a + b = a^-ab Dodo {a + bf-4(a Taco DSu +b' =(a _ L . L2 + b" =(a + b)^-3ab>(a + b)<0<^0 -508031. 11 Do do gia tri nho nhk cua A la -508031, dat dugc khi t = 504 hay x = 504 V.Taco ^ = - 4 ( x ' - ; c + l) + 3 | 2 x - l | = - ( 2 x - l ) ' + 3 | 2 x - l | - 3 = -2x-lp+3|2x-l|-3. Dat/ = |2x-l| v i - l < x < l nen ^e(l;3).Khid6 3 1 3^' 3 3 ' A = -t +3t-3 = - t - - — < — . Dau ' =' xay ra khi t = — hay x- —, 4 4 ^ 2 ^ 4 vi -1 < X < 1. Vay gia tri Ion nh^t ciia A la — . 4 8. Dat x = a + l,Z) = j + l,c = z + l. Taco a,b,ce[-2;2], a + b + c = 0. Ta can chung minh + 6^ + < 8. Vi trong 2 s6 3 s6 a,b,c c6 hai s6 cung d4u, giasu 6,c>0. Khi do +b^ 6-6. -^^' ^ x'-x +l Lof gidi. Ta CO x^ - x +1 > 0, Vx e R nen bat phuomg trinh tucmg duomg voi 2 x ^ + m x - 4 > - 6 ( x ^ - x + l), V X G R <::>8x^ + ( w - 6 ) x + 2>0, VxeR o A = (m - 6)^ - 64 < 0 -2 < w < 14. Vay-2x. 4. Tim m de bat phuong trinh sau dung voi Vx e R (m-l)x^ +{m + \)x + m-\>0. 5. Tim m de bat phuong trinh sau c6 tap nghiem la M .^^'^ ^ 3x^ -mx + 5 ^ 1<— <6. lx^-x + \ Huang dan giai bai tap phan 1.2 x ' - ( w + 2)x + 2m<0 (1) 1. Ki hieu + ( 7 w + 7)x + 7m<0 (2) Taco A, = ( w - 2 ) ^ > 0 ; A 2 = ( m - 7 ) ^ > 0 . Ta CO phuong trinh Phuong trinh x^ + - (m + 2) x + 2w = 0 luon c6 nghiem x = m hoac x = 2. + 7) x + 7m = 0 luon c6 nghiem x = -m hoac x = - 7 . Voi m = 2 hoac voi m = 7 thi he v6 nghiem. Voi m^2\m^l, m>0 nghiem cua ( l ) la 5, =(w;2) hoac S^-[2•,m)•, 'nghiem cua (2) la ^2 = ( - m ; - 7 ) hoac ^3 = ( - 7 ; - w ) . Ro rang 5, f l ^ j = 0 nen he v6 nghiem. Voi m<0 nghiem cua ( l ) la 5, = ( w ; 2 ) , nghiem cua (2) la -{-l;-m). 16 Ta CO iS", n a<-l a>4 Neu a<-l thi ox^ - 4 x + a - 3 < 0 , V x e M . Tirdo x+1 x + l>ax^ - 4x + a - 3 , V x e M <=>ax^-5x + a - 4 < 0 , V x e M <=>A = 2 5 - 4 a ( a - 4 ) < 0 (vi a<-l) \a<-l - V 4-V4T <=>S , <=>a< [4a'-16a-25>0 2 N6u a > 4 thi ox^ - 4x + a - 3 > 0, Vx e M. Tirdo :0 x+1 • 0 , V x e M O A = 2 5 - 4 a ( a - 4 ) < 0 ( v i a>4) <=> 4 + V4T a>4 4a'-16fl-25>0 Ket hop lai a e -Qo; 4-741 u 4+V4T -; + oo 3. Taco / ( / ( x ) ) >x<=>/(/(x))-x>0 Taco /(/W)-^ = /(/W)-/W+/W-^ i / ( x ) + 1 - (xL+.flx+-i-W-/fx)-x-~':":7:^7,'i '/'{x)-x'] ^(x^ +(a-\)x + a[f{x)-x] =[f{x)-x][f{x) + x + a] el + \)[x' +{a + \)x + a + 2). Bat g{x) = x'+{a-l)x + lc6 A^=a'-2a-3 h(x) = x^+{a + \)x + a + 2 CO A^=a^-2a-7. Do 3 < a < | nen tudo ^ ^ , d o d 6 h{x) = x^ +{a+ \)x + a + 2>0,\fxeR, f(f{x))-x>0 <=>g(x)>0<=>x^ + ( a - l ) x + l > O o l-a-Va'-2a-3 2 x> l-a + V a ' - 2 a - 3 fm-l>0 4.Tac6 ( w - l ) x ^ +(m + l)x + m-l>0,\/xGR<^ A<0 m-l>0 /w-l>0 / w l > 0 m<•ow>3. 3 (m + 1)2 - 4 ( w - l ) <0 [-3m'+10/n-3<0 m>3 5. Ta CO 2x^ - X +1 > 0, Vx G M. Do do bat phirong trinh tuong duong voi 3x^ - mx + 5 1< -,Vxe: x^ + ( l - m ) x + 4>0,VxGlS 2x'-x + l 9x'+(m-6)x + l>0,Vxe 3x -tnx + 5 < 6 , V x e : [ 2x'-x4-l A,=(l-m)'-16<0 {-3 0 dk dua wh phuong trinh bac hai theo t. 3.2. Dang (x + a)"^ + (x + Z?)'* = c. Cdch gidi: Dat x + = t se thu dugc phuong trinh trung phuong theo /. 3.3. Dangnghich dao ox'* + bx^ + cx^ ±bx + a ^ 0, a ^0. Cdch gidi: Chia ca hai ve cua phuong trinh cho n x^ 9^ 0 thu dugc phuong trinh + b x± a 1 = / (hoac x- \ — -t), bieu dien x^ + theo t, thay vao phuong Dat X + — V X X X trinh tren se thu dugc phuong trinh bac hai theo t. 3.4. Dang h6i quy ax"^ + bx^ + cx'^ + dx + e = 0, voi — = Kb a Cdch gidi: Gia su — = k^0. Chia ca hai vg cua phuong trinh cho x^ ^0 thu b dugc phuong trinh aa x ^ + ^ + 6 x + — + c = 0. I xj k ~ -) k~ Dat x + —= bieu dien x + ^ theo t, thay vao phuong trinh tren se thu X x^ dugc phuong trinh bac hai theo t. 3.5. Dang (x + a)(x +fe)(x+ c)(x + c/) = e, voi a + ^x^ +{a + d)x + ad^{x^ +{b + c)x + bc^ = e D?lt x^ +(a + d)x = r thu dugc phuong trinh bac hai theo /. 19 3.6. Dang (x + a)(x + b)(x + c)(x + d)-ex ,\(n ad = be Cdch giai: Viet phuong trinh duai dang ((x + aXx + cl)){(x + b){x + c)) = ex^ <=>(x^ + ( a + (i)x + fl - Xem thêm -