不等式公式小结
不等式的证明证律及重要公式证证重ab+222222ab2ab,ab()1、;可直接用,+???a+b+c?ab+bc+ca要2公
式22a+ba+b2+??ab?(a,b?R)、;要证明,会21122+ab
333、即可,3a+b+c?3abc(a+b+c?0
abc++3+3、~abc()~4(a,b,c?R)a+b+c?3?abc?3
|a|?|b|?|a+b|?|a|+|b|(a,b,c?R)~、5
证明方法
方法一,作差比证法,
1222 已知,~求证,a+b+c?。a+b+c=13
1的代证112222222 证,左,右(3a+3b+3c?1)====[3a+3b+3c?(a+b+c)]=33
1222=[(a?b)+(b?c)+(c?a)]?03
2a2b2cb+cc+aa+b+方法二,作上比证法~证a、b、c~且~求证,a?b?cabc>abc?R
abc222左abcabca?ba?cb?cb?ac?ac?ba?bb?cc?aaabbcc===()?()?() 证,bccaab+++右bcaabc
aaa?bab 当证?>1,?>0?()>1a>b>0bb
aaa?bab 当证??(0,1)?<0?()>10
1()>1()>1a>ba0,b>0a+b=1
112514422(a)(b) ?a+b? ?+++?8ab2
2222ABABABAB++++2 证?由公式,得,()???2222
442211+++ababab22244 ()[()]??=?a+b?222168
222A+BA+B(A+B)222 证?由?()?A+B?222
+1111ab11222 ? 左?[(a+)+(b+)]=[a+b+]=(1+) ;,*2ab2ab2ab
ab+112ab ? ?()=??4ab24
1252(14) ? ?+=(*)22
(n+1)(n+2)方法四,放证法, log>log(n>1)n(n1)+
(n+1) ? ~ ? n>1log>0n
n(n+2)log?log<1 ? 只要证, 即可(n1)(n1)++
11n(n+2)2n(n+2)2 左[(log+log)]=[log]< n1n1(n1)+++22
2211(n+2n+1)2(n+1)2 [(log]=[log]=1< n1(n1)++22
+(a+b)(a+b)?aa+bb方法五,分析法,证a~a~b~b~求证,(自证)1212?R11221212
nna+ba+bna?0,b?0方法六,证证猜想、证证法,证数学~求证,;自证,()?22