设实数x1,x2,x3,x4,x5均不小于1,且x1•x2•x3•x4•x5=729,则max{x1x2,x2x3,x3x4,x4x5}的最小值是______.-数学

题目简介

设实数x1,x2,x3,x4,x5均不小于1,且x1•x2•x3•x4•x5=729,则max{x1x2,x2x3,x3x4,x4x5}的最小值是______.-数学

题目详情

设实数x1,x2,x3,x4,x5均不小于1,且x1•x2•x3•x4•x5=729,则max{x1x2,x2x3,x3x4,x4x5}的最小值是______.
题型:填空题难度:中档来源:南通二模

答案

∵x1x2+x3x4≥2
class="stub"729
x5
,即取定一个x5后,x1x2,x3x4不会都小于
class="stub"729
x5

同样x2x3+x4x5≥2
class="stub"729
x1

class="stub"729
x5
+
class="stub"729
x1
≥2
class="stub"729×729
x1x5

使三个不等式等号都成立,则
x1x2=x3x4=
class="stub"729
x5

x2x3=x4x5=
class="stub"729
x1

x1=x5
即x1=x3=x5,x2=x4 x1x2=x2x3=x3x4=x4x5
所以729=x13×x22=
(x1x2)3
x2
,(x1x2)3=729×x2
x2最小为1,
所以x1x2最小值为9,
此时x1=x3=x5=9 x2=x4=1.
故答案为:9.

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