Bài tập Đại số 10 Nâng cao Phần 2 - Nguyễn Huy Đoan (chủ biên)

phucmg IV BAT DANG THl/C VA BAT PHaONG TRJNH A NHONG KIEN THQC CAN NHO Tinh chat cua bat dang thurc l)a>bvab>c=>a>c 2)a>boa + c>b + c 3) Ne'u c> thi a > b '^ ac> be Ne'u c < thi a > b ac < be Cdc he qua 4)a>bvac>d=>a a + c>b a>b- + c>b + d c 5)a>b>0vac>d>0=^ac>bd 6) a > ^ > va n G N* => a" > &" l)a>b>0=>Ja>^fb S)a>b=>^>^2 Bat dang thiirc ve gia trj tuyet doi Ddi vdi hai sd a, b y, ta cd ^ Id - \b\ , /7 > , t a e d a + b ^ i—r a + b r-r ^ > y/ab ; ^ = ^Jab 0, b>0,c >0,taed a + b + c ^ 3r-r- a + b + c ->/—— r > y/abc ; ~ = yjabc a = b = c Ap dung 1) Ne'u hai sd duong cd t6ng khdng doi thi tich cua ehiing Idn nha't hai sd dd bang 2) Ne'u hai sd duong cd tich khdng ddi thi tdng ciia chiing nhd nha't hai sd dd bang Bie'n doi tUdng duang cac bat phudng trinh Cho ba't phuong trinh fix) < gix) cd tap xac dinh ®, y = hix) la mdt ham sd xac dinh tren y^ Khi dd, tren 2), ba't phuong trinh fix) < gix) tuong duong vdi mdi ba't phuong trinh 1) fix) + hix) < gix) + hix) ; 2) fix)hix) < gix)hix) ndu hix) > vdi mpi x e S); 3) fix)hix) > gix)hix) ndu hix) < vdi mpi x e 3) Bat phUdng trinh va he bat phuong trinh bcic nhat mdt an • Giai va bien luan ba't phuong trinh ax + b thi niia mat phang (khdng kd bd id)) khdng ehda didm M la midn nghiem ciia (1) ^ 9 Chii y Ddi vdi bat phuong trinh ax + by + c X2) Ndi each khae, « / U ) < X e (x^ ; -^2)' afix) > O X > XT \a>0 J 2) \/x s R, ax + bx + c> c^ X < Xj [A < a \a ft thi — > ft ft + c a c 4.4 Cho «, ft, c, d la bdn sd duong va — < — Chdng minh rang : , a +b c +d a) — r — < — — , , (3 + ft f + J b) > 4.5 Cho ft, ti la hai sd duong va — < — Chihig minh rang a 'b^ a +c c b + d'^'d' 4.6 Cho a, ft, c, d la bdn sd duong Chiing minh rang 1< a b e d ~ 4.24 Cho a, ft, c la ba sd duong Tim gia tri- nho nha't ciia A= a b c ft + c + c + a + a + b 4.25 Trdn mat phing toa dp Oxy, ve dudng tron tam cd ban ki'nh R iR> 0) Tren cae tia Ox va Oy l^n lupt la'y hai didm A va B cho dudng thing AB ludn tie'p xiic vdi dudng tron dd Hay xae dinh toa dp eiia A va dd tam giac OAB cd dien tich nhd nha't 105 §2 DAI C i r O N G Vfi B A T PHLfONG TRINH 4.26 Trong eac menh dd sau, menh dd nao diing, menh dd nao sai, vi ? a) la mdt nghiem ciia ba't phuong trinh x^ + x + > b) - khdng la nghiem cua ba't phuong trinh x^ - 3x - < c) a la mdt nghiem ciia ba't phuong trinh x + (1 + a)x - a + < 4.27 Cac cap ba't phuong trinh sau cd tuong duong khdng, vi ? a) 2x - > va 2x - + x-2 > X- ' b) 2x - > va 2x - + ——- > x+2 x+2 ' c) X - < va x^(x - 3) < ; d) x - > va x^(x - 3) > ; e) X - > va (x - 2)^ > ; g) x - > va (x - 5)(x^ - 2x + 2) > 4.28, Tim didu kien xac dinh roi suy tap nghiem cua mdi bat phuong trinh sau : a) V x - > V - X ; b) V2x - < + V2x - ; e) d)3x+^—>2+ x-2 -^ six-3 < ^ ; Vx-3 ' ^ x-2 4.29 Khong giai bat phuong trinh hay giai thi'eh tai cac bat phuong trinh sau vo nghiem : a) V ^ + < ; b) (x - 1)^ + x^ < -3 ; e) x^ + (x - 3)^ + > (x - 3)^ + X- + ; d) Vl + 2(x + l)^ + > / l - x + x^ < 4.30 Khong giai bat phuong trinh, hay giai thich tai cae ba't phuong trinh sau nghiem diing vdi mpi x : a) ,v* + x^ + > ; c) X^ + (X - 1)^ + — > x^ X- + 106 b) ^^^f^ > ; x^ + 4.31 Tim didu kien xdc dinh ciia cac ba't phuong trinh sau : 1 ^ ' • a) (X +1)2T + X - n > ;' ^^ v G m 1 b) VTTT , + (X - — — > 2)(x - 3) x-4 4.32 Di giai baj: phuong trinh Vx - > V2x - (1), ban Nam da lam nhu sau : Do hai ve' eiia ba^t phuong trinh (1) luon khong am nen (1) tuong duong vdi (Vx-2)2 > (V2x - 3)2 hay X - > 2x - Do dd X < vay tap nghiem ciia (1) la (-QO, 1) Theo em, ban Nam giai da dung chua, vi ? 4.33 Ban Minh giai ba't phuong trinh , < (1) nhu sau : Vx2-2x-3 '^ + (l)x + 5< Vx^ - 2x - (X + 5)2 < x^ - 2x - 12x + 28 ; d) ^2x - - V x - l > 107 4.36 Giai cac he bit phuong trinh sau va bidu didn tap nghiem tren true sd : 3x + - < X + a) 6x-3 b) < 2x + 1; 4x + < 2x + > X - 7x - 4.37 Giai va bien luan cac bat phuong trinh (in x): a) mix -m)>0 ; b)(x- l)m>x + 2; ^ X - ab X - ac x - be + x + l Chia hai ve eho Vx + > , ta ed Vx - - > Vx + Vi X > nen Vx - < Vx + 1, dd V x - l - < Vx + vay ba't phuong trinh (1) vd nghiem Theo em, ban Nam giai diing hay sai, vi ? 4.39 Tim cac gia tri ciia m dd he bit phuong trinh sau ed nghiem : Ix + 4m^ < 2mx + l3x + > x - I 4.40 Tim cae gia tri eua m de he bat phuong trtnh sau vd nghiem : \mx + ; c) x(x + 5) < 2( x2 + 2) ; ^^ ^^x^ _ ^^^ _ io5 < ; d) 2( x + 2) ^ - 3,5 > 2x ; e) -X - 3x + 6< 4.60 Giai cac bit phuong trinh : 2x - a) -^ < ; x2 - 6x - x-3 ^ 2x - c) — >—... ^Ix - Vl4x - 49 = Vl4 ; c) I2V2IXI-I - | = ; d) X + V l - x ^ l = -V2(2x2 - 1) 4.77 Giai cae bit phuong trinh sau : a) V-x2 - x - > x + ; c) V2-X + 4x - >2 ; b) ^5x2 + 61x < 4x + ; 4.3x^ -3 4.78... x - l - V x - > V x - ; b) 2x(x - 1) + > 4x^ -x + ; 4 .101 Tim cac gia tri x thoa man : a) 1x2 - 2x - 3| - > |2x - l| ; b) 2| x + l| < |x - 2| + 3x + ; c) | V x - - l| + |Vx + - l| > ; d) jx -