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 will analyze the components of the function step by step.

### Step 1: Break down the function
The function can be rewritten as:
\[ f(x) = [5x + 7] + \cos(\pi x) - 5x \]
We will analyze each part of the function separately.

### Step 2: Analyze the greatest integer function
The greatest integer function \([5x + 7]\) can be expressed as:
\[ [5x + 7] = 5x + 7 - \{5x + 7\} \]
where \(\{x\}\) denotes the fractional part of \(x\). Thus, we can rewrite \(f(x)\):
\[ f(x) = (5x + 7 - \{5x + 7\}) + \cos(\pi x) - 5x \]
This simplifies to:
\[ f(x) = 7 - \{5x + 7\} + \cos(\pi x) \]

### Step 3: Determine the period of each component
1. **Period of the fractional part function \(\{5x + 7\}\)**:
- The fractional part function \(\{x\}\) has a period of 1. Therefore, \(\{5x + 7\}\) will have a period of:
\[ \frac{1}{5} \]

2. **Period of the cosine function \(\cos(\pi x)\)**:
- The cosine function \(\cos(x)\) has a period of \(2\pi\). Thus, \(\cos(\pi x)\) will have a period of:
\[ \frac{2\pi}{\pi} = 2 \]

### Step 4: Find the least common multiple (LCM) of the periods
Now, we need to find the LCM of the two periods:
- Period of \(\{5x + 7\}\) is \(\frac{1}{5}\)
- Period of \(\cos(\pi x)\) is \(2\)

To find the LCM, we can express \(2\) in terms of a fraction:
\[ 2 = \frac{10}{5} \]
Now, we can find the LCM of \(\frac{1}{5}\) and \(\frac{10}{5}\):
- The LCM of the numerators (1 and 10) is \(10\).
- The common denominator is \(5\).

Thus, the LCM is:
\[ \frac{10}{5} = 2 \]

### Conclusion
The period of the function \( f(x) = [5x + 7] + \cos(\pi x) - 5x \) is:
\[ \boxed{2} \]