已知a,b,c∈R+,a+b+c=1,求证:1a+1b+1c≥9.-数学

题目简介

已知a,b,c∈R+,a+b+c=1,求证:1a+1b+1c≥9.-数学

题目详情

已知a,b,c∈R+,a+b+c=1,求证:
1
a
+
1
b
+
1
c
≥9
题型:解答题难度:中档来源:不详

答案

证明:由题意知class="stub"1
a
+class="stub"1
b
+class="stub"1
c
=class="stub"a+b+c
a
+class="stub"a+b+c
b
+class="stub"a+b+c
c

=3+(class="stub"b
a
+class="stub"a
b
)+(class="stub"c
a
+class="stub"a
c
)+(class="stub"b
c
+class="stub"c
b

class="stub"b
a
+class="stub"a
b
≥2
class="stub"c
a
+class="stub"a
c
≥2
class="stub"b
c
+class="stub"c
b
≥2

当且仅当a=b=c时,取等号,
class="stub"1
a
+class="stub"1
b
+class="stub"1
c
≥9

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