Question

数学解答题

已知：x满足不等式2(log½X)²+7log½X+3≤0
试求：f(x)=log2 x/2 * log2 x/4 的最大值和最小值。

Answer 1

解：
2(log½X)²+7log½X+3≤0
[log(1/2)(x)+3][2log(1/2)(x)+1]≤0
-3≤log(1/2)(x)≤-1/2
√2≤x≤8
1/2≤log2(x)≤3

f(x)=log2(x/2)log2(x/4)
=[log2(x)-1][log2(x)-2]
=[log2(x)]^2-3log2(x)+2
=[log2(x)-3/2]^2-1/4

log2(x)=3/2时，有f(x)min=-1/4
log2(x)=3时，有f(x)max=2

Answer 2

-3<=log½X<=1/2 根号2/2<=X<=8

f(x)=log2 x/2 * log2 x/4 =（log2x-1）（log2x-2）

令log2x=t -1/2<=t<=3

y=t^2-3t+2 最大值t=-1/2 15/4

最小值t=3/2 1/2