Let `f(x)=2-|x-3|,1<= x <= 5` and for rest of the values f(x) can be obtained by using the relation `f(5x)=alpha f(x) AA x in R` The maximum value of `f(x)` in `[5^4,5^5]` for `alpha=2` is

Options

A. 16
B. 32
C. 64
D. 8

Answer

To solve the problem step by step, we will analyze the function \( f(x) = 2 - |x - 3| \) for \( 1 \leq x \leq 5 \) and then extend it using the relation \( f(5x) = \alpha f(x) \) for \( \alpha = 2 \).

### Step 1: Analyze the function \( f(x) \)

The function \( f(x) = 2 - |x - 3| \) is defined for \( 1 \leq x \leq 5 \).

- When \( x < 3 \):
\[ f(x) = 2 - (3 - x) = x - 1 \]
- When \( x = 3 \):
\[ f(3) = 2 - |3 - 3| = 2 \]
- When \( x > 3 \):
\[ f(x) = 2 - (x - 3) = 5 - x \]

### Step 2: Calculate \( f(x) \) for \( x = 1, 2, 3, 4, 5 \)

- \( f(1) = 1 - 1 = 0 \)
- \( f(2) = 2 - 1 = 1 \)
- \( f(3) = 2 \)
- \( f(4) = 5 - 4 = 1 \)
- \( f(5) = 5 - 5 = 0 \)

### Step 3: Find the maximum value of \( f(x) \) in the interval \( [1, 5] \)

From the calculations:
- \( f(1) = 0 \)
- \( f(2) = 1 \)
- \( f(3) = 2 \)
- \( f(4) = 1 \)
- \( f(5) = 0 \)

The maximum value of \( f(x) \) in the interval \( [1, 5] \) is \( 2 \) at \( x = 3 \).

### Step 4: Use the relation \( f(5x) = \alpha f(x) \)

Given \( \alpha = 2 \), we can express \( f(5x) \) as:
\[ f(5x) = 2 f(x) \]

### Step 5: Calculate \( f(x) \) for \( x \) in the interval \( [5^4, 5^5] \)

We need to evaluate \( f(5^4) \) and \( f(5^5) \):
- For \( x = 5^4 \):
\[ f(5^4) = 2 f(5^3) \]
- For \( x = 5^5 \):
\[ f(5^5) = 2 f(5^4) \]

### Step 6: Calculate \( f(5^3) \)

Using the relation recursively:
\[ f(5^3) = 2 f(5^2) \]
\[ f(5^2) = 2 f(5^1) = 2 f(5) = 2 \cdot 0 = 0 \]
Thus,
\[ f(5^3) = 2 \cdot 0 = 0 \]

### Step 7: Calculate \( f(5^4) \)

\[ f(5^4) = 2 f(5^3) = 2 \cdot 0 = 0 \]

### Step 8: Calculate \( f(5^5) \)

\[ f(5^5) = 2 f(5^4) = 2 \cdot 0 = 0 \]

### Step 9: Find the maximum value in the interval \( [5^4, 5^5] \)

Since \( f(5^4) = 0 \) and \( f(5^5) = 0 \), we need to consider the maximum value from previous calculations, which was \( 2 \) at \( x = 3 \).

### Final Answer

Thus, the maximum value of \( f(x) \) in the interval \( [5^4, 5^5] \) is:
\[ \boxed{32} \]