Find the Range of function
f(x) = [|sin x| + |cosx |] , where [.] denotes are greatest integer function , is :

Options

A. {0}
B. `{0,1}`
C. `{1}`
D. None

Answer

To find the range of the function \( f(x) = \lfloor |\sin x| + |\cos x| \rfloor \), where \( \lfloor . \rfloor \) denotes the greatest integer function, we can follow these steps:

### Step 1: Analyze the function \( |\sin x| + |\cos x| \)

The first step is to understand the behavior of \( |\sin x| + |\cos x| \). Since both \( \sin x \) and \( \cos x \) oscillate between -1 and 1, their absolute values will oscillate between 0 and 1. Therefore, \( |\sin x| + |\cos x| \) will range from 0 to 2.

### Step 2: Determine the maximum value of \( |\sin x| + |\cos x| \)

To find the maximum value of \( |\sin x| + |\cos x| \), we can use the Cauchy-Schwarz inequality or trigonometric identities.

Using the identity:
\[ |\sin x| + |\cos x| = \sqrt{(\sin^2 x + \cos^2 x) + 2|\sin x \cos x|} = \sqrt{1 + 2|\sin x \cos x|} \]
The maximum value occurs when \( |\sin x \cos x| \) is maximized. The maximum value of \( |\sin x \cos x| \) is \( \frac{1}{2} \), which occurs at \( x = \frac{\pi}{4} + n\pi \) for integers \( n \).

Thus, the maximum value of \( |\sin x| + |\cos x| \) is:
\[ \sqrt{1 + 2 \cdot \frac{1}{2}} = \sqrt{2} \]

### Step 3: Find the range of \( |\sin x| + |\cos x| \)

Now we need to find the minimum value of \( |\sin x| + |\cos x| \). The minimum occurs when either \( \sin x \) or \( \cos x \) is 0. For example, at \( x = 0 \) or \( x = \frac{\pi}{2} \), we have:
\[ |\sin 0| + |\cos 0| = 0 + 1 = 1 \]
\[ |\sin \frac{\pi}{2}| + |\cos \frac{\pi}{2}| = 1 + 0 = 1 \]

Thus, the minimum value of \( |\sin x| + |\cos x| \) is 1.

### Step 4: Determine the range of \( f(x) \)

Now we know that:
- The minimum value of \( |\sin x| + |\cos x| \) is 1.
- The maximum value of \( |\sin x| + |\cos x| \) is \( \sqrt{2} \).

Since \( \sqrt{2} \) is approximately 1.414, the values of \( |\sin x| + |\cos x| \) will range from 1 to \( \sqrt{2} \).

### Step 5: Apply the greatest integer function

Now, we apply the greatest integer function \( \lfloor . \rfloor \) to the range of \( |\sin x| + |\cos x| \):
- The values of \( |\sin x| + |\cos x| \) range from 1 to \( \sqrt{2} \).
- Therefore, \( \lfloor |\sin x| + |\cos x| \rfloor \) will take values from \( \lfloor 1 \rfloor \) to \( \lfloor \sqrt{2} \rfloor \).

Since \( \lfloor \sqrt{2} \rfloor = 1 \), the only integer value that \( f(x) \) can take is 1.

### Conclusion

Thus, the range of the function \( f(x) = \lfloor |\sin x| + |\cos x| \rfloor \) is:
\[ \{1\} \]