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定义区间 (a,b),[a,b),(a,b],[a,b](a, b),[a, b),(a, b],[a, b] 的长度均为 d=b−ad=b-a ，多个区间并集的长度为各区间长度之和，例如， $(1,2) \...
Mar 8, 2024
定义区间 (a,b),[a,b),(a,b],[a,b](a, b),[a, b),(a, b],[a, b] 的长度均为 d=b−ad=b-a ，多个区间并集的长度为各区间长度之和，例如， (1,2)∪[3,5)(1,2) \cup[3,5) 的长度 d=(2−1)+(5−3)=3d=(2-1)+(5-3)=3. 用 [x][x] 表示不超过 xx 旳最大整数，记 {x}=x−[x]\{x\}=x-[x], 其中 x∈Rx \in \mathbf{R}. 设 f(x)=[x]⋅{x},g(x)=12x−1f(x)=[x] \cdot\{x\}, g(x)=\frac{1}{2} x-1 ，当 −2≤x≤k-2 \leq x \leq k 时，不等式 f(x)<g(x) 解集的区间长度为 107105\frac{107}{105} ，则实数 kk 的最小值为（）。 A. 307\frac{30}{7} B. 163\frac{16}{3} C. 6 D. 7
Generated Graph
Solution by Steps
step 2
From the asksia-ll calculation, we know that the length of intervals where f(x) < g(x) is 107105 \frac{107}{105} when x=17110 x = \frac{171}{10} or approximately 17.1 17.1
step 3
To find the minimum k k , we compare the value of x x where the inequality changes with the given options. The closest value less than 17.1 17.1 that is an option is 6 6
step 4
Therefore, the minimum value of k k that satisfies the condition is 6 6
C
Key Concept
Interval Length Calculation
Explanation
To find the minimum k k , we use the length of intervals where f(x) < g(x) and compare it with the given options to find the closest value that does not exceed the calculated x x .
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