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函数

f(x)

的定义域为

\mathrm{R}

, 满足

f(x+1)=2 f(x)

，且当

x \in(0,1]

时， $f(x)=4 x(x...

Mar 8, 2024

函数

f(x)

的定义域为

\mathrm{R}

, 满足

f(x+1)=2 f(x)

，且当

x \in(0,1]

时，

f(x)=4 x(x-1)

，若对任意的

x \in(-\infty, m]

，都有

f(x) \geq-3

，则

m

的取值范围是

Solution by Steps

step 1

Define the function

f(x)

for

x \in (0,1]

as given:

f(x) = 4x(x-1)

step 2

Find the value of

f(x)

at the endpoints of the interval

(0,1]

f(0)

and

f(1)

step 3

Calculate

f(0)

using the definition of

f(x)

f(0) = 4 \cdot 0 \cdot (0-1) = 0

step 4

Calculate

f(1)

using the definition of

f(x)

f(1) = 4 \cdot 1 \cdot (1-1) = 0

step 5

Determine the critical points of

f(x)

in the interval

(0,1]

by setting the derivative

f'(x)

equal to zero and solving for

x

step 6

Compute the derivative:

f'(x) = 4(2x-1)

step 7

Set the derivative equal to zero:

4(2x-1) = 0

step 8

Solve for

x

x = \frac{1}{2}

step 9

Evaluate

f(x)

x = \frac{1}{2}

f\left(\frac{1}{2}\right) = 4 \cdot \frac{1}{2} \cdot \left(\frac{1}{2}-1\right) = -1

step 10

Since

f(x) \geq -3

for

x \in (0,1]

, and the minimum value of

f(x)

in this interval is

-1

, the condition is satisfied for

x \in (0,1]

step 11

Use the functional equation

f(x+1) = 2f(x)

to extend the definition of

f(x)

x \in (-\infty, 0]

step 12

Find the largest value of

m

such that

f(x) \geq -3

for all

x \in (-\infty, m]

step 13

Since

f(x)

is defined recursively, check values of

f(x)

for

x \in (-\infty, 0]

using the functional equation

step 14

For

x = 0

f(x) = 0 \geq -3

. For

x = -1

f(-1) = \frac{f(0)}{2} = 0 \geq -3

. Continue checking values of

x

until f(x) < -3

step 15

Since

f(x)

is decreasing for x < 0, find the first

x

such that f(x) < -3

step 16

The value of

m

is the largest

x

for which

f(x) \geq -3

Answer

The value of

m

is the largest

x

for which

f(x) \geq -3

Key Concept

Recursive Function and Interval Analysis

Explanation

The key concept involves understanding the recursive nature of the function

f(x)

and analyzing its behavior over different intervals to find the maximum value of

m

such that

f(x) \geq -3

for all

x \in (-\infty, m]