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8. 已知函数

f(x)

的定义域为

\mathrm{R}, f(x)>f(x-1)+f(x-2)

, 且当

x<3

时,

f(x)=x

,则下列...

Aug 14, 2024

Generated Graph

Solution by Steps

step 2

For

x = 0, 1, 2

, we have

f(0) = 0

f(1) = 1

, and

f(2) = 2

. We can calculate

f(3)

using the inequality: f(3) > f(2) + f(1) = 2 + 1 = 3. Thus, f(3) > 3

step 3

Continuing this process, we find f(4) > f(3) + f(2) > 3 + 2 = 5, so f(4) > 5. Similarly, we can derive that f(5) > f(4) + f(3) > 5 + 3 = 8, and so on

step 4

This pattern suggests that

f(n)

grows faster than the Fibonacci sequence, which implies that

f(10)

and

f(20)

will be significantly larger than 100 and 1000, respectively

step 5

Therefore, we can conclude that f(10) > 100 and f(20) > 1000, making options A and B correct. However, we need to determine which statements are definitely true based on the growth pattern

Key Concept

Inequalities and function growth rates

Explanation

The function

f(x)

grows faster than linear due to the recursive inequality, leading to values exceeding 100 and 1000 for larger inputs.