The value of `k(k >0)` such that the length of the longest interval in which the function `f(x)=sin^(-1)|sink x|+cos^(-1)(cosk x)` is constant is `pi/4` is/ are 8 (b) 4 (c) 12 (d) 16

Options

A. 8
B. 4
C. 12
D. 16

Explanation

To solve the problem, we need to find the value of \( k \) such that the function

\[ f(x) = \sin^{-1} |\sin(kx)| + \cos^{-1} (\cos(kx)) \]

is constant over the interval of length \( \frac{\pi}{4} \).

### Step 1: Analyze the Components of the Function

The function consists of two parts:
1. \( \sin^{-1} |\sin(kx)| \)
2. \( \cos^{-1} (\cos(kx)) \)

### Step 2: Determine the Range of Each Component

1. **For \( \sin^{-1} |\sin(kx)| \)**:
- The function \( |\sin(kx)| \) oscillates between 0 and 1.
- Therefore, \( \sin^{-1} |\sin(kx)| \) will vary between 0 and \( \frac{\pi}{2} \).

2. **For \( \cos^{-1} (\cos(kx)) \)**:
- The function \( \cos(kx) \) oscillates between -1 and 1.
- Thus, \( \cos^{-1} (\cos(kx)) \) will vary between 0 and \( \pi \).

### Step 3: Combine the Two Components

The function \( f(x) \) can be expressed as:

\[ f(x) = \sin^{-1} |\sin(kx)| + \cos^{-1} (\cos(kx)) \]

### Step 4: Set the Function to be Constant

For \( f(x) \) to be constant, the sum of the two components must equal a constant value. We want this constant value to equal \( \frac{\pi}{4} \).

### Step 5: Identify the Conditions for Constancy

The function \( f(x) \) will be constant if both components do not change over the interval of \( x \). This occurs when:

- \( kx \) is restricted to a specific range where both \( |\sin(kx)| \) and \( \cos(kx) \) do not change.

### Step 6: Determine the Length of the Interval

We need to find the interval length \( \frac{\pi}{4} \).

- The periodicity of \( \sin(kx) \) is \( \frac{2\pi}{k} \).
- The periodicity of \( \cos(kx) \) is also \( \frac{2\pi}{k} \).

Thus, for \( f(x) \) to be constant over an interval of length \( \frac{\pi}{4} \), we can set:

\[ \frac{2\pi}{k} = \frac{\pi}{4} \]

### Step 7: Solve for \( k \)

To find \( k \):

\[ 2\pi = \frac{\pi}{4} k \]

Multiplying both sides by 4:

\[ 8\pi = \pi k \]

Dividing both sides by \( \pi \):

\[ k = 8 \]

### Conclusion

The value of \( k \) such that the length of the longest interval in which the function \( f(x) \) is constant is \( \frac{\pi}{4} \) is:

\[ \boxed{8} \]