若-3≤log½ X(以½为底数X的对数)≤-½,求f(x)=(log2 x/2)·(log2 x/4)的最大值和最小值.(以2为低x/2的对数乘以2为低x/4的对数)

问题描述:

若-3≤log½ X(以½为底数X的对数)≤-½,求f(x)=(log2 x/2)·(log2 x/4)的最大值和最小值.(以2为低x/2的对数乘以2为低x/4的对数)

f(x) = [log2 (x/2)]·[log2(x/4)]
= (log2 x - 1)·(log2 x - 2)
= [(log2 x) - (3/2)]^2 - (1/4)
而,-3《log(1/2) x《-1/2 可化简为:1/2《log2 x《3
∴-1《[(log2 x) - (3/2)]《3/2
∴0《[(log2 x) - (3/2)]^2《 9/4
∴-1/4《[(log2 x) - (3/2)]《2
即:f(x)最大值2 ,最小值-1/4