若实数a,b,c满足2a+2b=2a+b,2a+2b+2c=2a+b+c,则c的最大值是______.-数学

题目简介

若实数a,b,c满足2a+2b=2a+b,2a+2b+2c=2a+b+c,则c的最大值是______.-数学

题目详情

若实数a,b,c满足2a+2b=2a+b,2a+2b+2c=2a+b+c,则c的最大值是______.
题型:填空题难度:中档来源:重庆

答案

由基本不等式得2a+2b≥2
2a2b
=2×2class="stub"a+b
2
,即2a+b≥2
2a2b
=2×2class="stub"a+b
2
,所以2a+b≥4,
令t=2a+b,由2a+2b+2c=2a+b+c可得2a+b+2c=2a+b2c,所以2c=class="stub"t
t-1
=1+class="stub"1
t-1

因为t≥4,所以1<class="stub"t
t-1
≤class="stub"4
3
,即1<2c≤class="stub"4
3
,所以0<c≤log2class="stub"4
3
=2-log23

故答案为:2-log23

更多内容推荐