已知a,b,c属于R*,且a+b+c=1,求证1/(a+b)+1/(b+c)+1/(c+a)大于等于9/2 用均值不等式做
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已知a,b,c属于R*,且a+b+c=1,求证1/(a+b)+1/(b+c)+1/(c+a)大于等于9/2 用均值不等式做
已知a,b,c属于R*,且a+b+c=1,求证1/(a+b)+1/(b+c)+1/(c+a)大于等于9/2 用均值不等式做
已知a,b,c属于R*,且a+b+c=1,求证1/(a+b)+1/(b+c)+1/(c+a)大于等于9/2 用均值不等式做
∵2/(a+b)+2/(b+c)+2/(c+a)
=[(a+b)+(b+c)+(c+a)]/(a+b)
+[(b+c)+(a+b)+(c+a)]/(b+c)
+[(c+a)+(a+b)+(b+c)]/(c+a)
=1+(b+c)/(a+b)+(c+a)/(a+b)
+1+(a+b)/(b+c)+(c+a)/(b+c)
+1+(a+b)/(c+a)+(b+c)/(c+a)
=3+[(b+c)/(a+b)+(a+b)/(b+c)]
+[(c+a)/(a+b)+(a+b)/(c+a)]
+[(c+a)/(b+c)+(b+c)/(c+a)]
而(b+c)/(a+b)+(a+b)/(b+c)≥2,
(c+a)/(a+b)+(a+b)/(c+a)≥2,
(c+a)/(b+c)+(b+c)/(c+a)≥2,
∴2/(a+b)+2/(b+c)+2/(c+a)≥3+2+2+2=9,
∴1/(a+b)+1/(b+c)+1/(c+a)≥9/2.
注:本题用柯西不等式证明较简明.
建议你用柯西不等式,
[1/(a+b)+1/(b+c)+1/(c+a)]×[a+b+b+c+c+a]≥﹙1+1+1﹚²=9
1/(a+b)+1/(b+c)+1/(c+a)≥9/2