f(x)=(log2 X/4)*(log2 X/2),x∈[1/2,4]的值域

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f(x)=(log2 X/4)*(log2 X/2),x∈[1/2,4]的值域
f(x)=(log2 X/4)*(log2 X/2),x∈[1/2,4]的值域

f(x)=(log2 X/4)*(log2 X/2),x∈[1/2,4]的值域
f(x)=(log2 X/4)*(log2 X/2)
=(log2 X/4)*(log2 (X/4)*2)
=(log2 X/4)*(log2 (X/4)+log2 2)
=(log2 X/4)*(log2 (X/4)+1)
=(log2 X/4)^2+log2 (X/4)
=((log2 X/4)+1/2)^2-1/4
x∈[1/2,4],x/4∈[1/8,1]
因为x/4<=1,所以log2 X/4为减函数
所以当log2 X/4=-1/2,即x=2√2时有最小值-1/4
当x=1/2,log2 X/4=-3时,有最大值6
f(x)的值域为[-1/4,6]