Which of the following function are not periodic (where [.] denotes greatest integer function:
`f(x)=[sin 3x]+|cos 6x|`

Explanation

To determine whether the function \( f(x) = [\sin 3x] + |\cos 6x| \) is periodic or not, we will analyze the periodicity of each component of the function.

### Step 1: Identify the Period of Each Component

1. **Period of \( \sin 3x \)**:
- The standard period of \( \sin x \) is \( 2\pi \).
- For \( \sin 3x \), the period is given by:
\[ T_1 = \frac{2\pi}{3} \]

2. **Period of \( |\cos 6x| \)**:
- The standard period of \( \cos x \) is \( 2\pi \).
- For \( \cos 6x \), the period is given by:
\[ T_2 = \frac{2\pi}{6} = \frac{\pi}{3} \]
- However, since we are considering the absolute value \( |\cos 6x| \), the period is halved:
\[ T_2 = \frac{\pi}{6} \]

### Step 2: Determine the Period of the Entire Function

- The function \( f(x) \) is a sum of two periodic functions: \( [\sin 3x] \) and \( |\cos 6x| \).
- The periodicity of \( f(x) \) will depend on the least common multiple (LCM) of the periods of the two components.

### Step 3: Find the LCM of the Periods

- The periods we have are:
- \( T_1 = \frac{2\pi}{3} \)
- \( T_2 = \frac{\pi}{6} \)

To find the LCM, we can express both periods with a common denominator:

- \( T_1 = \frac{2\pi}{3} = \frac{4\pi}{6} \)
- \( T_2 = \frac{\pi}{6} \)

Now, the LCM of \( \frac{4\pi}{6} \) and \( \frac{\pi}{6} \) is:
\[ \text{LCM}\left(\frac{4\pi}{6}, \frac{\pi}{6}\right) = \frac{4\pi}{6} = \frac{2\pi}{3} \]

### Step 4: Analyze the Greatest Integer Function

- The greatest integer function \( [\sin 3x] \) is not periodic because it takes on integer values based on the output of \( \sin 3x \).
- The function \( [\sin 3x] \) will change its value whenever \( \sin 3x \) crosses an integer, which does not happen at regular intervals.

### Conclusion

Since \( [\sin 3x] \) is not periodic, the entire function \( f(x) = [\sin 3x] + |\cos 6x| \) is also not periodic.

### Final Answer
Thus, the function \( f(x) = [\sin 3x] + |\cos 6x| \) is **not periodic**.

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