数学求函数y=log2(x/4)log4(x/2) (2小于等于x小于等于4）的值域
y=log2(x/4)log4(x/2)
=(log2 x -log2 4)(log4 x -log4 2)
=(log2 x -2 )(1/2 log2 x -1/2)
=1/2(log2 x -2)(log2 x -1)
=1/2〔(log2 x)^2 -3log2 x +2〕
=1/2(log2 x -3/2)^2 -1/8
2≤x≤4, 令log2 x=t , 1≤t≤2
y=1/2(log2 x -3/2)^2 -1/8
=1/2(t -3/2)^2 -1/8
函数图象为抛物线,开口向上,
当t=3/2时,y有最小值=-1/8,此时x=2√2
当x=2,或者4时,y=0
所以,...全部
y=log2(x/4)log4(x/2)
=(log2 x -log2 4)(log4 x -log4 2)
=(log2 x -2 )(1/2 log2 x -1/2)
=1/2(log2 x -2)(log2 x -1)
=1/2〔(log2 x)^2 -3log2 x +2〕
=1/2(log2 x -3/2)^2 -1/8
2≤x≤4, 令log2 x=t , 1≤t≤2
y=1/2(log2 x -3/2)^2 -1/8
=1/2(t -3/2)^2 -1/8
函数图象为抛物线,开口向上,
当t=3/2时,y有最小值=-1/8,此时x=2√2
当x=2,或者4时,y=0
所以,值宇为:-1/8≤y≤0
。
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