Chuang U fifll f HUDTIG TRlnH Bat ding thuc va bat phuong trinh la nhung khai niem ma chung ta da lam quen lop duoi Chuong se hoan thien hon cac khai niem do, dong thoi cung cap cho chiing ta nhung ki^n thuc mdi nhu van de xet dau cua nhi thiic bac nhat va dau cCia tam thuc bac hai Chung co nhieu ung dung quan trong viec giiii va bien lu^n cac phuong trinh va bat phuong trinh Chung ta can nam vung cac kien thuc do, dong thdi ren luyen kl nang ap dung chiing de giai cac bai toan khuon kho ciHa chuong tnnh 103 t /^_ BAT D A N G T H l / G V A C H U N G M I N H B X T D A N G THtTC On tap va bo sung tmh chat cua biit dang thurc Gia sir a vab la hai sd thuc Cac menh dd "a > b", "a < b", "a > b'\ "a < b dugc ggi la nhung bdt dang thdc Ciing nhu cac menh dd Idgic khac, mdt bdt dang thiic cd thd ddng hoac sai Chiing minh mdt bdt ddng thiic la chiing minh bdt dang thiic dd diing Dudi day la mdt sd tinh chdt da bidt ciia bdt dang thtfc a>b va b>c ^^ a>c a>b a + o b + c Ndu c > thi a > Z? «:> ac>be Ndu cbb va c>d => a + c>b + d; a + oh b-c; a>b>0 va c>d>0 ^^ aobd; a>b>0 va neN* => a">b"; a>b>0 yfa > 4b ; a>b-^ \fa > ^fb Vi du Khdng dung bang sd hoac may tinh, hay so sanh hai sd >/2 + >/3 vd Gidi Gia suf V2 + \/3 < Do hai vd ciia bdt dang thiic dd ddu dugng ndn V2 + >^ ( x - l ) o x ^ > x - o x ^ - x + l + l>0 (x-l)^ + l > Hidn nhidn (x - 1) + > vdi mgi x nen ta cd bdt dang thiic cdn chiing minh n Vi du Chiing minh rang ndu a, b, c la dd dai ba canh cua mdt tam giac thi ib + c - a)ic + a- b)ia + b - c) < abc Gidi Ta cd cac bdt dang thiic hidn nhidn sau : 2 a > a -ib-c) =ia-b + c)ia + b-c) b^ >b^-ic-af = ib-c + a)ib + c-a) c^ >c^-ia-bf = ic-a + b)ic + a- b) Do a, b, c la dd ddi ba canh cua mdt tam giac ndn tdt ca cdc vd ciia cac bdt dang thiic tren ddu duong Nhan cac vd tuong ling cua ba bdt dang thiic tren, ta duoc a V c ^ >ib + c- afic + a- hfia + b-cf ' Ldy can bdc hai cua hai vd, ta dugc bdt dang thiic cdn chiing minh D Bat d^ng thurc ve gia tri tuyet doi Tur dinh nghia gid tti ttiyet ddi, ta suy cac tinh chdt sau day a\< a< \a\ vdi mgi a G E |x 0) - 105 Sau day la hai bdt ddng thiic quan ttgng khdc vd gia tri tuydt ddi (vidt dudi dang bdt dang thiic kep) a\ -\b\2 +2 +2 =6 Vft a ^c b ^ a c a H£ QUA Niu hai so duang thay ddi nhung cd tdng khdng dd'i thi tich ciia chiing ldn nhdt vd chi hai sddd bdng Niu hai so duang thay ddi nhung cd tich khdng ddi thi tdng ciia chung nhd nhdt vd chi hai sddd bdng Chicng minh Gia sir hai sd duong x vd y cd tdng x + y = S khdng ddi Khi dd, —= — > yfxy nen xy < — Dang thiic xay vd chi x = y Do dd, tich xy dat gid tri ldn nhdt bang — va chi klii x = y 107 Gia sir hai sd duang x va y cd tich xy = P khdng ddi Khi dd ^ ^ ^ > 4xy = yfP nen x + y > 2>/P, Dang thiic xay va chi x = y Do dd, tdng x + y dat gid tri nhd nhdt bdiig v P Idii va chi x = y • TJNGDIJNG Trong tdt cd cdc hinh cha nhdt cd cung chu vi, hinh vudng cd dien tich Ian nhdt Trong tdt cd cdc hinh ,chii nhdt cd cung diin tich, hinh vudng co chu vi nhd nhdt Vi du Tim gid tri nho nhdt cua ham s6fix) = x H— vdi x > X Gidi Do X > nen ta cd fix) = x + — > Jx.— = 2yi3 va X V X fix) = 2V3 M) = 2v3 b) Doi vdi ba sd Ichdng dm Ta da bidt flf + ft + C la trung binh cdng cua ba sd a, ft, c Ta ggi Vflftc la trung binh nhan cua ba sd dd Ngudi ta ciing chiing minh dugc kdt qua tuong tu dinh Ii tren cho trudng hgp ba sd khdng am Vdi mgi a > 0, ft > 0, c > 0, ta cd Dang thiic xay va chi a = b = c Ndi each khdc, trung binh cdng cOa ba sd khdng dm ldn han hodc bdng trung binh nhdn cua chiing Trung binh cdng ciia ba sd'khdng dm bdng trung binh nhdn cua chiing vd chikhi ba sddd bdng 108 Vi du Chiing minh rang ndu a,ft,c la ba sd duong thi 1 ia + b + c) — + - + - >9 a b cj Khi ndo xay dang thiic ? Gidi Vi a,b,c la ba sd duong nen a + b + c > ¥abc (dang thiic xay va chi a =ft= c) va i,i4>33-i-' dang thiic xay va chi — = — = — a b c Do dd V abc a b c I I I (a +ft+ c) —+ - + - >3Vaftc.3 / — =9 a b c \ abc a =b=c Dang thiic xay va chi 1-1-1 a b c Vdy dang thiic xay Idii va chi a = b = c a H3| Phat biSu kd't qua tuong tuhd qud d phin a) cho trudng hgp ba so duang Cau hoi va bdi tap Oiling minh rang, ndu a > ft va aft > thi — < — a ft Chting minh rdng nira chu vi cua mdt tam gidc ldn hon dd ddi mdi canh cua tam gidc dd Chiing minh rang a^ + b^ + c^ > ab + be + ca vdi mgi sd thuc a, ft, c Ding thiic xay va chi a = ft = c Hay so sdnh cdc kdt qua sau day : a) V2000 + V2005 vd V2002 + V2003 (khdng diing bang sd hoac may tinh); b) yla + + yJa + A va yfa + yfa + ia > 0) 109 1 Chiing minh rdng, ndu a > v a f t > t h i — + — > a ft a + b Chiing minh rdng, neu a > va ft > thi a^ + ft^ > abia + ft) Dang thiic xay nao ? a) Chiing minh rdng a^ + aft + ft^ > vdi mgi sd thuc a, ft b) Chiing minh rang vdi hai sd thuc a, ft y, ta cd a^ +b^ > a^b + ab^ Chiing minh rdng, ndu a, ft, c la dd dai cdc canh cua mdt tam gidc thi a^ +b^ +c^ < 2(aft + bc + ca) Chiing minh rang, ndu a > vd ft > thi a +b a^ +b^ 10 a) Chiing minh rang, ndu x > y > thi a^ +b^ ^ >^ y 1+x 1+y b) Chiing minh rdng ddi vdi hai sd y a, ft, ta cd -^—!—< l + |a-ft| 1^1 ^ I I l + |a| l + |ft| 11 Chiing minh rang : a) Ndu a, ft la hai sd ciing ddu thi —+— > ; ft a b) Ndu a, ft la hai sd trdi ddu thi - + - < - ft a 12 Tun gid tri ldn nhdt va gia tti nhd nhdt ciia ham sd fix) = (x + 3)(5-x) vdi -3