If `f(x) = sin^(-1)(m^(2) -3m+ 1)sec (absx)/3-[e^(m-4)](sin{x})`, where {} denotes fractional part of x. If f(x) if f(x) is periodic, then number of possible integral values of m is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine the number of possible integral values of \( m \) for which the function

\[ f(x) = \sin^{-1}(m^2 - 3m + 1) \sec\left(\frac{|x|}{3}\right) - e^{m-4} \sin\{x\} \]

is periodic, we will follow these steps:

### Step 1: Determine the range of \( \sin^{-1}(m^2 - 3m + 1) \)

The function \( \sin^{-1}(y) \) is defined for \( y \) in the range \([-1, 1]\). Therefore, we need to ensure that:

\[ -1 \leq m^2 - 3m + 1 \leq 1 \]

### Step 2: Solve the inequalities

#### Inequality 1: \( m^2 - 3m + 1 \geq -1 \)

This simplifies to:

\[ m^2 - 3m + 2 \geq 0 \]

Factoring gives:

\[ (m-1)(m-2) \geq 0 \]

The critical points are \( m = 1 \) and \( m = 2 \). The solution to this inequality is:

\[ m \leq 1 \quad \text{or} \quad m \geq 2 \]

#### Inequality 2: \( m^2 - 3m + 1 \leq 1 \)

This simplifies to:

\[ m^2 - 3m \leq 0 \]

Factoring gives:

\[ m(m-3) \leq 0 \]

The critical points are \( m = 0 \) and \( m = 3 \). The solution to this inequality is:

\[ 0 \leq m \leq 3 \]

### Step 3: Find the intersection of the two ranges

From the first inequality, we have \( m \leq 1 \) or \( m \geq 2 \). From the second inequality, we have \( 0 \leq m \leq 3 \).

The intersection of these two ranges is:

- For \( m \leq 1 \): \( 0 \leq m \leq 1 \)
- For \( m \geq 2 \): \( 2 \leq m \leq 3 \)

Thus, the combined range for \( m \) is:

\[ 0 \leq m \leq 1 \quad \text{and} \quad 2 \leq m \leq 3 \]

### Step 4: Identify integral values of \( m \)

The integral values of \( m \) in the ranges \( [0, 1] \) and \( [2, 3] \) are:

- From \( [0, 1] \): \( m = 0, 1 \)
- From \( [2, 3] \): \( m = 2, 3 \)

Thus, the possible integral values of \( m \) are \( 0, 1, 2, 3 \).

### Step 5: Conclusion

The total number of possible integral values of \( m \) is:

\[ \text{Total integral values} = 4 \]

### Final Answer

The number of possible integral values of \( m \) is \( 4 \).

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