已知x、y、z均为正数,求证:33(1/x+1/y+1/z)≤1/x2+1/y2+1/z2.
问题描述:
已知x、y、z均为正数,求证:
(
3
3
+1 x
+1 y
)≤1 z
.
+1 x2
+1 y2
1 z2
答
证明:由柯西不等式得(12+12+12)(
+1 x2
+1 y2
)≥(1 z2
+1 x
+1 y
)2…(5分)1 z
则
×
3
≥
+1 x2
+1 y2
1 z2
+1 x
+1 y
,1 z
即
(
3
3
+1 x
+1 y
)≤1 z
…(10分)
+1 x2
+1 y2
1 z2