Period of the function `f(x) =(1)/(3){sin 3x + |sin 3x | + [sin 3x]}` is (where [.] denotes the greatest integer function )

Options

A. `(pi)/(3)`
B. `( 2pi )/(3)`
C. `( 4pi )/(3)`
D. `pi`

Explanation

To find the period of the function \( f(x) = \frac{1}{3} \left( \sin(3x) + |\sin(3x)| + [\sin(3x)] \right) \), we need to analyze the periodicity of each component of the function.

### Step 1: Determine the period of \( \sin(3x) \)
The standard sine function \( \sin(x) \) has a period of \( 2\pi \). Therefore, the function \( \sin(3x) \) will have a period given by:

\[ \text{Period of } \sin(3x) = \frac{2\pi}{3} \]

**Hint:** Remember that the period of \( \sin(kx) \) is \( \frac{2\pi}{k} \).

### Step 2: Determine the period of \( |\sin(3x)| \)
The absolute value function \( |\sin(x)| \) has a period of \( \pi \), since it repeats every half cycle of the sine wave. Therefore, the period of \( |\sin(3x)| \) is:

\[ \text{Period of } |\sin(3x)| = \frac{\pi}{3} \]

**Hint:** The absolute value of a sine function \( |\sin(kx)| \) has a period of \( \frac{\pi}{k} \).

### Step 3: Determine the period of \( [\sin(3x)] \)
The greatest integer function \( [x] \) (or floor function) takes the value of \( x \) and rounds it down to the nearest integer. The function \( \sin(3x) \) oscillates between -1 and 1, and thus \( [\sin(3x)] \) will take values -1, 0, or 1. The period of \( [\sin(3x)] \) is the same as that of \( \sin(3x) \):

\[ \text{Period of } [\sin(3x)] = \frac{2\pi}{3} \]

**Hint:** The greatest integer function inherits the periodicity of the function it is applied to, but may change the output values.

### Step 4: Find the least common multiple (LCM) of the periods
Now we need to find the least common multiple of the periods we have calculated:

1. Period of \( \sin(3x) = \frac{2\pi}{3} \)
2. Period of \( |\sin(3x)| = \frac{\pi}{3} \)
3. Period of \( [\sin(3x)] = \frac{2\pi}{3} \)

The LCM of these periods can be calculated as follows:

- The LCM of \( \frac{2\pi}{3} \) and \( \frac{\pi}{3} \) is \( \frac{2\pi}{3} \) since \( \frac{2\pi}{3} \) is already a multiple of \( \frac{\pi}{3} \).

Thus, the overall period of the function \( f(x) \) is:

\[ \text{Period of } f(x) = \frac{2\pi}{3} \]

### Final Answer
The period of the function \( f(x) \) is \( \frac{2\pi}{3} \).

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