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)
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![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)](/uploads/image/z/10343160-0-0.jpg?t=%E8%8B%A5-3%E2%89%A4log%26frac12%3B+X%EF%BC%88%E4%BB%A5%26frac12%3B%E4%B8%BA%E5%BA%95%E6%95%B0X%E7%9A%84%E5%AF%B9%E6%95%B0%EF%BC%89%E2%89%A4-%26frac12%3B%2C%E6%B1%82f%EF%BC%88x%EF%BC%89%3D%EF%BC%88log2+x%2F2%EF%BC%89%C2%B7%EF%BC%88log2+x%2F4%EF%BC%89%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC.%EF%BC%88%E4%BB%A52%E4%B8%BA%E4%BD%8Ex%2F2%E7%9A%84%E5%AF%B9%E6%95%B0%E4%B9%98%E4%BB%A52%E4%B8%BA%E4%BD%8Ex%2F4%E7%9A%84%E5%AF%B9%E6%95%B0%EF%BC%89)
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)
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)
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
因为f(x)=(log2 x/2)·(log2 x/4)所以f(x)=xlog2(½)·xlog2(¼)
=-xlog2(2)·-xlog2(4)=(-x)·(-2x)=2x²
有因为3≤log½ X≤-½,所以 根号2≤x≤8
因为函数在2到8单调递增所以f(x)最大值为2·64=128最小值为2·2=4<...
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因为f(x)=(log2 x/2)·(log2 x/4)所以f(x)=xlog2(½)·xlog2(¼)
=-xlog2(2)·-xlog2(4)=(-x)·(-2x)=2x²
有因为3≤log½ X≤-½,所以 根号2≤x≤8
因为函数在2到8单调递增所以f(x)最大值为2·64=128最小值为2·2=4
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