Question

（高一）若x满足2(log(1/2)x)^2-14log(4)x+3≤0,求f(x)=[log(2)(x/2)]*{log(√2)[(√x)/2]}的最大值和最小

若x满足2(log(1/2)x)^2-14log(4)x+3≤0,求f(x)=[log(2)(x/2)]*{log(√2)[(√x)/2]}的最大值和最小值，并求此时x的值。
rt

Answer 1

2(log(1/2)x)^2-14log(4)x+3≤0
2(log(2)x）^2-7log(2)x+3≤0
=> 1/2≤log(2)x≤2
log(2)√2≤log(2)x≤log(2)8
∴√2≤x≤8
f(x)=2log(2)(x/2)-log(2)[(√x)/2]
又∵f(x)在[√2,8]上递增
∴f(x)min=f(√2)=3/4
f(x)max=f(8)=2