If `f(x)= cos [(pi^(2))/(2)] x + sin[(-pi^(2))/(2)]x,[x]` denoting the greatest integer function,then

Options

A. `f(0) = 0`
B. `f((pi)/(3)) = sqrt((3)-1)/(2)`
C. `f((pi)/(2)) = -1`
D. `f(pi) = 0`

Explanation

To solve the problem, we need to analyze the function \( f(x) = \cos\left(\frac{\pi^2}{2} x\right) + \sin\left(-\frac{\pi^2}{2} x\right) \) where \([x]\) denotes the greatest integer function.

### Step 1: Calculate \( \frac{\pi^2}{2} \)
First, we need to find the value of \( \frac{\pi^2}{2} \).

Using \( \pi \approx 3.14 \):
\[ \pi^2 \approx (3.14)^2 \approx 9.8596 \]
Thus,
\[ \frac{\pi^2}{2} \approx \frac{9.8596}{2} \approx 4.9298 \]

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

### Step 3: Rewrite the function
Now we can rewrite the function using the greatest integer value:
\[ f(x) = \cos(4x) + \sin(-4x) \]
Since \( \sin(-\theta) = -\sin(\theta) \), we can simplify this to:
\[ f(x) = \cos(4x) - \sin(4x) \]

### Step 4: Evaluate \( f(0) \)
Now, let's evaluate \( f(0) \):
\[ f(0) = \cos(4 \cdot 0) - \sin(4 \cdot 0) = \cos(0) - \sin(0) = 1 - 0 = 1 \]

### Step 5: Evaluate \( f\left(\frac{\pi}{3}\right) \)
Next, we evaluate \( f\left(\frac{\pi}{3}\right) \):
\[ f\left(\frac{\pi}{3}\right) = \cos\left(4 \cdot \frac{\pi}{3}\right) - \sin\left(4 \cdot \frac{\pi}{3}\right) \]
Calculating \( 4 \cdot \frac{\pi}{3} = \frac{4\pi}{3} \):
\[ f\left(\frac{\pi}{3}\right) = \cos\left(\frac{4\pi}{3}\right) - \sin\left(\frac{4\pi}{3}\right) \]
Using the unit circle:
\[ \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}, \quad \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]
Thus,
\[ f\left(\frac{\pi}{3}\right) = -\frac{1}{2} - \left(-\frac{\sqrt{3}}{2}\right) = -\frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{\sqrt{3} - 1}{2} \]

### Step 6: Evaluate \( f\left(\frac{\pi}{2}\right) \)
Next, we evaluate \( f\left(\frac{\pi}{2}\right) \):
\[ f\left(\frac{\pi}{2}\right) = \cos\left(4 \cdot \frac{\pi}{2}\right) - \sin\left(4 \cdot \frac{\pi}{2}\right) = \cos(2\pi) - \sin(2\pi) = 1 - 0 = 1 \]

### Step 7: Evaluate \( f(\pi) \)
Finally, we evaluate \( f(\pi) \):
\[ f(\pi) = \cos(4\pi) - \sin(4\pi) = 1 - 0 = 1 \]

### Summary of Results
- \( f(0) = 1 \)
- \( f\left(\frac{\pi}{3}\right) = \frac{\sqrt{3} - 1}{2} \)
- \( f\left(\frac{\pi}{2}\right) = 1 \)
- \( f(\pi) = 1 \)

### Conclusion
The correct options based on the evaluations are \( f(0) = 1 \), \( f\left(\frac{\pi}{2}\right) = 1 \), and \( f(\pi) = 1 \).