Let f(x) = {x} , For f (x) , x = 5 is (where {*} denotes the fractional part)

Options

A. a point of local maxima
B. a point of local minima
C. neither a point of local minima nor maxima
D. None of these

Answer

To solve the problem, we need to analyze the function \( f(x) = \{x\} \), where \( \{x\} \) denotes the fractional part of \( x \). The fractional part of \( x \) is defined as:

\[ \{x\} = x - \lfloor x \rfloor \]

where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \).

### Step 1: Evaluate \( f(5) \)

To find \( f(5) \):

\[ f(5) = \{5\} = 5 - \lfloor 5 \rfloor = 5 - 5 = 0 \]

### Step 2: Analyze \( f(x) \) around \( x = 5 \)

Next, we need to analyze the behavior of \( f(x) \) as \( x \) approaches 5 from both sides:

1. **For \( x \) approaching 5 from the left (\( x = 5 - \epsilon \), where \( \epsilon \) is a small positive number)**:
\[ f(5 - \epsilon) = \{5 - \epsilon\} = (5 - \epsilon) - \lfloor 5 - \epsilon \rfloor = (5 - \epsilon) - 4 = 1 - \epsilon \]

2. **For \( x \) approaching 5 from the right (\( x = 5 + \epsilon \))**:
\[ f(5 + \epsilon) = \{5 + \epsilon\} = (5 + \epsilon) - \lfloor 5 + \epsilon \rfloor = (5 + \epsilon) - 5 = \epsilon \]

### Step 3: Compare values around \( x = 5 \)

Now we compare the values:

- As \( x \) approaches 5 from the left:
\[ f(5 - \epsilon) = 1 - \epsilon \quad \text{(which approaches 1 as } \epsilon \to 0\text{)} \]

- As \( x \) approaches 5 from the right:
\[ f(5 + \epsilon) = \epsilon \quad \text{(which approaches 0 as } \epsilon \to 0\text{)} \]

### Step 4: Determine local maxima or minima

From the analysis:

- \( f(5 - \epsilon) > f(5) \) for values approaching from the left (1 > 0).
- \( f(5 + \epsilon) < f(5) \) for values approaching from the right (0 < 0).

This indicates that \( f(5) = 0 \) is a local minimum because the function value decreases to the left and increases to the right.

### Conclusion

Thus, the answer to the question is that at \( x = 5 \), \( f(x) \) has a point of local minima.

### Final Answer

The correct option is: **Point of local minima**.

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