If `f(x) = cos [pi]x + cos [pi x]`, where `[y]` is the greatest integer function of y then `f(pi/2)` is equal to

Options

A. cos 3
B. 0
C. cos 4
D. cos 5

Answer

To find the value of \( f\left(\frac{\pi}{2}\right) \) for the function \( f(x) = \cos[\pi x] + \cos[\pi x] \), where \([y]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Substitute \( x = \frac{\pi}{2} \) into the function
We start with the function:
\[ f(x) = \cos[\pi x] + \cos[\pi x] \]
Now, substituting \( x = \frac{\pi}{2} \):
\[ f\left(\frac{\pi}{2}\right) = \cos\left[\pi \cdot \frac{\pi}{2}\right] + \cos\left[\pi \cdot \frac{\pi}{2}\right] \]

### Step 2: Calculate \( \pi \cdot \frac{\pi}{2} \)
Calculating the expression inside the greatest integer function:
\[ \pi \cdot \frac{\pi}{2} = \frac{\pi^2}{2} \]
Using the approximate value of \( \pi \approx 3.14 \):
\[ \frac{\pi^2}{2} \approx \frac{(3.14)^2}{2} = \frac{9.8596}{2} \approx 4.9298 \]

### Step 3: Apply the greatest integer function
Now we apply the greatest integer function:
\[ \left[\frac{\pi^2}{2}\right] = [4.9298] = 4 \]

### Step 4: Substitute back into the function
Now we can substitute this back into our function:
\[ f\left(\frac{\pi}{2}\right) = \cos(4) + \cos(4) \]
This simplifies to:
\[ f\left(\frac{\pi}{2}\right) = 2\cos(4) \]

### Final Result
Thus, the final value of \( f\left(\frac{\pi}{2}\right) \) is:
\[ f\left(\frac{\pi}{2}\right) = 2\cos(4) \]