Let `f(x)={(0.1)^(3[x])}`. (where [.] denotes greatest integer function and denotes fractional part). If `f(x + T) =f(x) AA x in 0`, where T is a fixed positive number then the least x value of T is

Options

A. 2
B. 4
C. 6
D. None of these

Explanation

To solve the problem, we need to analyze the function given and find the least positive value of \( T \) such that \( f(x + T) = f(x) \) for all \( x \) in the interval \( [0, T) \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function is defined as:
\[ f(x) = (0.1)^{3[x]} \]
where \([x]\) is the greatest integer function (floor function). This means that \( f(x) \) takes the value of \( (0.1)^{3n} \) where \( n \) is the greatest integer less than or equal to \( x \).

2. **Identifying the Periodicity**:
For \( f(x + T) = f(x) \), the value of \([x + T]\) must equal \([x]\) for all \( x \) in the interval \( [0, T) \). This implies that \( T \) must be an integer, as the greatest integer function changes only at integer values.

3. **Finding the Condition**:
The condition \([x + T] = [x]\) holds true if \( x + T \) does not cross the next integer. This means:
\[ [x + T] = [x] \implies n \leq x < n + 1 \implies n \leq x + T < n + 1 \]
This leads to:
\[ n \leq x + T < n + 1 \implies -T < x < 1 - T \]
Since \( x \) is in the interval \([0, T)\), we need to ensure that \( T \) is such that \( 1 - T > 0 \) or \( T < 1 \).

4. **Finding the Least Positive Value of \( T \)**:
The smallest integer \( T \) that satisfies the condition \( T < 1 \) is \( T = 1 \). However, since \( T \) must be a fixed positive number, we need to check the periodicity condition for \( T = 1 \):
\[ f(x + 1) = f(x) \]
For \( x \) in the interval \([0, 1)\), \([x] = 0\), thus:
\[ f(x) = (0.1)^{3 \cdot 0} = 1 \]
and
\[ f(x + 1) = (0.1)^{3 \cdot 1} = 0.1^3 = 0.001 \]
This does not hold. Therefore, we need to check for \( T = 3 \):
\[ f(x + 3) = (0.1)^{3[x + 3]} = (0.1)^{3([x] + 3)} = (0.1)^{3[x] + 9} = (0.1)^{3[x]} \cdot (0.1)^9 = f(x) \]
This holds true.

5. **Conclusion**:
The least positive value of \( T \) such that \( f(x + T) = f(x) \) for all \( x \) in the interval \([0, T)\) is:
\[ \boxed{3} \]