: x2y2 z z (x+y+z)'
Trong BDT (l) cho x-y=7=l;,ft11 '3 Ta )
-{t:,-LJ-{ 4 -: s ,.5,-J-" ./ , !.--.r1 J " r 1 '.,1_ t I :r l- . ,?"'^+ \i LtrU w tArW -;$;'"or<-'*
di vi6t ndy dC c{p tltin mQt sd bdi to5n tim
,: i. ;.. , t
hang so tdt nhat trong bii torin bdt clang
thric (BDT). NQi dung cria n6 clugc ph6t bi6u
kh5 dcvn gi6n nhu sau: Tim hing sd k 16n nh|,t
(nh6 nh6t) dC mQt BDT lu6n dirng v6i mQt sd
gi6 thi6t ndo cl6 cua c6c bi6n. De gi6i dqng to6n ndy ta thudng thuc hiOn theo c6c bu6c sau:
Badc l. Chgn gi6 tri tlflc bi6t ctra c5c bitin hopc
,. A
drinh giri tryc ti6p cric bi€n dd chi ra cli6u kiQn
cdn ct.r. k.
Baoc 2. Chimg minh BDT dd cho ihing v6i gi6
tri cta k (lffinh6t, nhd ntr6t; vria tim dugc.
Vi6c chon chc gi6 tri dAc bi6t nhu o bu6c I
khdng phii ltc ndo cfrng d6 dang md cdn phu thu6c vdo d6u b[ng vd c6c di6u kiQn cria gi6 thi6t. Xin dua ra mOt sd Thi du minh hga:
Th{ dy l. Tim s6 thtrc k>0 ktn nhdt sao cho
bh dang thtrc sau dting voi mai s6 thac x,y,z:
x' +.v' +:a +-rv:(-r+.r,*z) ) k(xy + yz + xz)2 (l) Loi gidi.
I
Trong BDT (l) cho x-y=7=l;,ft11. '3 Ta) )
chimg minh BDT (l) dung khi k=:, hrc lA
cin chimg minh BDT
444)x- + y* + z* + xyz(x+ y + zl>:lxy + yz + xz1' 11.11 x- + y* + z* + xyz(x+ y + zl>:lxy + yz + xz1' 11.11 3' Ta c6 (1.1) e 3(xa + yo + zo1 > 2(r'y' + y' z' + x'r'1+ ryz(x + y + z) Ap dr,rng BDT a2 + b2 + c2 ) ab + bc + ca,
Va,b,c e IR. ta duoc:
2(xa + yo + zo)> 2(*'y' + y'r' + x'z'1 vit
4 4 4-
x* +y- +z') xyz(x+y+z), vAy BDT (1.2)) )
clugc chung minh. Do do max k = i .A
5
Th{ dy 2. Tim hdng si5 k > 0 ktn nhdt sao cho
bft dang thuc sau dung vcti moi a,b,c,d duong
ta-+h+ctf ' 'l 7a (o+h+c+c115 +2a (a+b+c+zfis)
> k.abcdr 127
> k.abcdr 127
Ta chimg minh BDT (2) dirng v1i k =31.24.5 ,
tuc ld c6n chimg minh:
7 = (a + b + c)l}o 1o + O + c + d|s + 2a (a + b + c + zflsf
> 37 .2o.5.abcd3 (2.t1. ta c6
r >lldfr .l l^ ll"[atc + a\' +zo (1"[aoc+ za)'] = s.
L'
CAn chimg minh S >3'.2o.s.abcd'
et^ .( * L\' *z'.( *4)' r r".r*.r.( -L )'--'(.-',,tffi ) ltffi ) -'''''lw I --'(.-',,tffi ) ltffi ) -'''''lw I
Ddt ' r = +. 1'labc Ta cAn chung minlr:
zo.(z +t)' +zo.(z +zt)' >36.20.5.t' (2.2)Ap dlrng BDT Cauchy ta co: Ap dlrng BDT Cauchy ta co:
3+t>-2^ht > (3+l)s >2t.3'.t'Ji
3 + 2t = 3 + t + t >31,bV = (3 + 2t)s >3u f 1,b' t .
= YT (2.2) > 3u 2o t' 12Ji + fi,m).
Do 2J-3t * i,l* = ,fi, - .1u -tdi r
r+vJl * : :
*'ffi *'ffi ,,
-1 -1
,lt,.f .* =s,n6n BDT (2.2) duqc chimg minh. Ddu 'o-" xiy n6n BDT (2.2) duqc chimg minh. Ddu 'o-" xiy
ra khi l:3. Vay maxk: 31.24.5 . J
NhQn xit.BDT (2.1) I|LBET thuAn nh6t v6i c6c
bi6n a,b,c,d n6n vi6c dit ' t =+ llabc d6 clua ve
mQt biiin h rat U nhi6n. MQt sO bei toan hrcmg t.u:
BAI TOAN
Ti}I HANG SOTOT NHAT
TRO}{C IATOiXC TI{U(
r,t xuAN u4r
(GV THPT chuyAn Wnh Ptuic