Period of the function `f(x) = [5x + 7] + cospix - 5x` where [·] denotes greatest integer function is

Options

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

Answer

To find the period of the function \( f(x) = [5x + 7] + \cos(\pi x) - 5x \), where \([ \cdot ]\) denotes the greatest integer function, we can break down the function into its components and analyze their periods.

### Step 1: Identify the components of the function

The function consists of three parts:
1. The greatest integer function \([5x + 7]\)
2. The cosine function \(\cos(\pi x)\)
3. The linear term \(-5x\)

### Step 2: Analyze the greatest integer function

The greatest integer function \([x]\) is a step function that has a period of 1. This is because it jumps at every integer value of \(x\).

### Step 3: Analyze the cosine function

The function \(\cos(\pi x)\) is periodic with a period of \(2\) because the general period of \(\cos(kx)\) is given by \(\frac{2\pi}{k}\). Here, \(k = \pi\), so the period is:
\[ \text{Period of } \cos(\pi x) = \frac{2\pi}{\pi} = 2 \]

### Step 4: Analyze the linear term

The term \(-5x\) is linear and does not have a period. It continuously decreases without repeating.

### Step 5: Combine the periods

To find the overall period of the function \(f(x)\), we need to consider the periodic components:
- The period of \([5x + 7]\) is \(1\).
- The period of \(\cos(\pi x)\) is \(2\).

Since the linear term \(-5x\) does not contribute to the period, we need to find the least common multiple (LCM) of the periods of the periodic functions:
\[ \text{LCM}(1, 2) = 2 \]

### Conclusion

Thus, the period of the function \(f(x) = [5x + 7] + \cos(\pi x) - 5x\) is \(2\).