题目

若-3大于等于log2分之1x小于等于-2分之1,求f(x)=(log以2为底2分之x为对数)乘（log以2为底4分之x）的最大值
和最小值

提问时间：2020-07-26

答案

-3 ≤ log(1/2) x ≤ - 1/2
log(1/2) x 单调减
(1/2)^(-1/2) ≤ x ≤ (1/2)^(-3)
x∈[根号2,8]
f(x) = [log(2) (x/2)] * [ log(2) (x/4)]
= [log(2) x - log(2) 2 ] * [ log(2) x - log(2) 4 ]
= [log(2) x - 1 ] * [ log(2) x - 2 ]
= [log(2) x]^2 - 3 log(2) x +2
= [log(2) x - 3/2 ]^2 - 9/4 + 2
= [log(2) x - 3/2 ]^2 - 1/4
x=2^(3/2)=2根号2时有极小值
最小值f(2根号2)= [log(2) 2根号2 - 3/2 ]^2 - 1/4 = （3/2-3/2)^2 - 1/4 = -1/4
x∈[根号2,2根号2)时单调减；x∈[2根号2,8)时单调增
f(根号2)= [log(2) 根号2 - 3/2 ]^2 - 1/4 = (1/2-3/2)^2-1/4 = 3/4
f(8)= [log(2) 8 - 3/2 ]^2 - 1/4 = (3-3/2)^2-1/4 = 2
∴最大值f(x)max=f(8)=2