If `f(x) = [2 + 5|n| sin x]`, where `n in I` has exactly 9 points of non-derivability in `(0, pi)`, then possible values of n are (where [x] dentoes greatest integer function)

Options

A. `+- 3`
B. `+- 2`
C. `+- 1`
D. None of these

Answer

To solve the problem, we need to analyze the function \( f(x) = \lfloor 2 + 5|n| \sin x \rfloor \) and determine the conditions under which it has exactly 9 points of non-derivability in the interval \( (0, \pi) \).

### Step 1: Understanding the Function
The function \( f(x) \) involves the greatest integer function (floor function) applied to \( 2 + 5|n| \sin x \). The term \( \sin x \) varies between 0 and 1 in the interval \( (0, \pi) \).

### Step 2: Finding the Range of \( f(x) \)
Since \( \sin x \) reaches its maximum value of 1 at \( x = \frac{\pi}{2} \), we can find the maximum and minimum values of \( f(x) \):
- Minimum value: \( f(x) = \lfloor 2 + 5|n| \cdot 0 \rfloor = \lfloor 2 \rfloor = 2 \)
- Maximum value: \( f(x) = \lfloor 2 + 5|n| \cdot 1 \rfloor = \lfloor 2 + 5|n| \rfloor \)

### Step 3: Identifying Points of Non-Differentiability
The function \( f(x) \) will be non-differentiable at points where \( 2 + 5|n| \sin x \) is an integer. We need to find the values of \( n \) such that the number of integers in the range \( [2, 2 + 5|n|] \) is exactly 9.

### Step 4: Setting Up the Equation
Let \( k = 2 + 5|n| \). The integers in the interval \( [2, k] \) are \( 2, 3, 4, \ldots, k \). The number of integers in this interval is given by:
\[ \text{Number of integers} = k - 2 + 1 = k - 1 \]
We want this to equal 9:
\[ k - 1 = 9 \implies k = 10 \]

### Step 5: Solving for \( n \)
Now we have:
\[ 2 + 5|n| = 10 \]
Subtracting 2 from both sides:
\[ 5|n| = 8 \]
Dividing by 5:
\[ |n| = \frac{8}{5} = 1.6 \]
Thus, the possible values of \( n \) are:
\[ n = \pm 1.6 \]

### Conclusion
The possible values of \( n \) such that \( f(x) \) has exactly 9 points of non-derivability in the interval \( (0, \pi) \) are \( n = 1.6 \) and \( n = -1.6 \).