The period of the function `f(x)= [6x+7]+cospix-6x ,` where `[dot]` denotes the greatest integer function is:

Options

A. 3
B. `2pi`
C. 2
D. None of these

Explanation

To find the period of the function \( f(x) = [6x + 7] + \cos(\pi x) - 6x \), where \([ \cdot ]\) denotes the greatest integer function, we will analyze the components of the function step by step.

### Step 1: Identify the components of the function
The function consists of three parts:
1. The greatest integer function: \([6x + 7]\)
2. The cosine function: \(\cos(\pi x)\)
3. The linear term: \(-6x\)

### Step 2: Determine the period of \(\cos(\pi x)\)
The cosine function \(\cos(kx)\) has a period of \(\frac{2\pi}{k}\). In our case, \(k = \pi\):
\[ \text{Period of } \cos(\pi x) = \frac{2\pi}{\pi} = 2 \]

### Step 3: Determine the period of the greatest integer function \([6x + 7]\)
The greatest integer function \([x]\) is a step function that has a period of 1. However, since we have \([6x + 7]\), we need to consider the effect of the coefficient 6:
\[ \text{Period of } [6x] = \frac{1}{6} \]
Thus, the period of \([6x + 7]\) is also \(\frac{1}{6}\).

### Step 4: Combine the periods
To find the overall period of the function \(f(x)\), we need to find the least common multiple (LCM) of the periods of the individual components:
- Period of \(\cos(\pi x) = 2\)
- Period of \([6x + 7] = \frac{1}{6}\)

To find the LCM of \(2\) and \(\frac{1}{6}\):
1. Convert \(2\) to a fraction: \(2 = \frac{12}{6}\)
2. Now we have \(\frac{12}{6}\) and \(\frac{1}{6}\).

The LCM of \(\frac{12}{6}\) and \(\frac{1}{6}\) is \(\frac{12}{6} = 2\).

### Final Answer
The period of the function \(f(x)\) is:
\[ \text{Period of } f(x) = 2 \]

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