函数y=log2底X\4* log4底x\2 【x大于等于2小于等于4】求该函数的值域
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![函数y=log2底X\4* log4底x\2 【x大于等于2小于等于4】求该函数的值域](/uploads/image/z/629407-55-7.jpg?t=%E5%87%BD%E6%95%B0y%3Dlog2%E5%BA%95X%5C4%2A+log4%E5%BA%95x%5C2+%E3%80%90x%E5%A4%A7%E4%BA%8E%E7%AD%89%E4%BA%8E2%E5%B0%8F%E4%BA%8E%E7%AD%89%E4%BA%8E4%E3%80%91%E6%B1%82%E8%AF%A5%E5%87%BD%E6%95%B0%E7%9A%84%E5%80%BC%E5%9F%9F)
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函数y=log2底X\4* log4底x\2 【x大于等于2小于等于4】求该函数的值域
函数y=log2底X\4* log4底x\2 【x大于等于2小于等于4】求该函数的值域
函数y=log2底X\4* log4底x\2 【x大于等于2小于等于4】求该函数的值域
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]