已知a,b,c∈R+,a+b+c=1,求证:1/a+1/b+1/c≥9.
问题描述:
已知a,b,c∈R+,a+b+c=1,求证:
+1 a
+1 b
≥9. 1 c
答
证明:由题意知
+1 a
+1 b
=1 c
+a+b+c a
+a+b+c b
a+b+c c
=3+(
+b a
)+(a b
+c a
)+(a c
+b c
)c b
∴
+b a
≥2,a b
+c a
≥2,a c
+b c
≥2.c b
当且仅当a=b=c时,取等号,
∴
+1 a
+1 b
≥9.1 c