已知x满足不等式2[log（1/2）x]^2+7log(1/2)x+3《0求函数f（x）=[log（2）x/4]*[log(2)x/2]的最大值和最小值.

Question

已知x满足不等式2[log（1/2）x]^2 +7log(1/2) x +3《0
求函数f（x）=[log（2）x/4]*[log(2) x/2]的最大值和最小值.

Answer 1

2[log（1/2）x]^2 +7log(1/2) x +3≤0
【2log(1/2) x+1】【log(1/2) x+3】≤0
得-3≤log(1/2) x≤-1/2
所以1/2≤log(2) x≤3 令a=log(2)x,则1/2≤a≤3
f（x）=[log（2）x/4]*[log(2) x/2]
=[log（2）x-log(2)4]*[log(2) x-log(2)2]
=[log（2）x-2]*[log(2) x-1]
所以f(a)=（a-2）（a-1）
=a²-3a+2
=(a-3/2)²-1/4
当a=3/2时,最小值为-1/4
当a=3时,最大值为2