若2{[log(1/2)x]^2}-14[log4(x)]+3≤0,求f(x)=log2(x/2)log√2[(√x)/2]的值域

Question

若2{[log(1/2)x]^2}-14[log4(x)]+3≤0,求f(x)=log2(x/2) log√2[(√x)/2]的值域

Answer 1

log(1//2)x=-log2(x) log4 (x)=1/2log2(x)
2【log2(x)]^2-7log2(x)+3≤0
∴1/2≤log2(x)≤3
∵f(x)=[log2(x)-1][log2(x)-2]
=[log2(x)]^2-3log2(x)+2
=[log2(x)-3/2]^2-1/4
∴当log2(x)=1/2、3/2、3时,f(x)=3/4、-1/4、2
∴-1/4≤f(x)≤2