题目

函数y=log2底X4* log4底x2 【x大于等于2小于等于4】求该函数的值域

提问时间：2020-10-11

答案

y=log2 x/4 ×log4 x/2=lg(x/4)/lg2 * lg(x/2)/lg4
=(lgx-lg4)/lg2 * (lgx-lg2)/(2lg2)
=(lgx-2lg2) * (lgx-lg2) / [2(lg2)^2]
=[(lgx)^2 - 3lg2 lgx +2(lg2)^2] / [2(lg2)^2]
=[(lgx - 3/2 lg2)^2 -1/4 (lg2)^2] / [2(lg2)^2]
当lgx - 3/2 lg2,即x=2^(3/2) = 2根号2时,有极小值ymin=(-1/4)/2 = -1/8
lgx单调增
2≤x＜2根号2时单调减；
2根号2＜x≤4时单调增.
x=2时,y=[(lg2 - 3/2 lg2)^2 -1/4 (lg2)^2] / [2(lg2)^2] = 0
x=4时,y=[(lg4 - 3/2 lg2)^2 -1/4 (lg2)^2] / [2(lg2)^2] = 0
∴值域[-1/8,0]