If `f(x)=cos[pi^(2)]x+cos[-pi^(2)]x`, where `[x]` stands for the greatest integer function, then

Options

A. `f((pi)/(2))=0`
B. `f(pi)=-1`
C. `f(-pi)=1`
D. `f((pi)/(4))=(1)/(sqrt(2))`

Answer

To solve the problem, we need to evaluate the function \( f(x) = \cos[\pi^2]x + \cos[-\pi^2]x \) where \([x]\) denotes the greatest integer function.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function is given as:
\[ f(x) = \cos[\pi^2]x + \cos[-\pi^2]x \]
We need to calculate the values of \( f(x) \) for specific inputs.

2. **Calculating \(\pi^2\)**:
We know that \( \pi \approx 3.14 \), thus:
\[ \pi^2 \approx 9.87 \]
Therefore, the greatest integer function \([\pi^2]\) will give us:
\[ [\pi^2] = 9 \]
Similarly, since \(-\pi^2\) is approximately \(-9.87\), we have:
\[ [-\pi^2] = -10 \]

3. **Substituting Values into the Function**:
Now we can rewrite \( f(x) \) as:
\[ f(x) = \cos(9x) + \cos(-10x) \]
Using the property of cosine, \(\cos(-\theta) = \cos(\theta)\), we can simplify this to:
\[ f(x) = \cos(9x) + \cos(10x) \]

4. **Evaluating \( f \) at Specific Points**:
We will evaluate \( f(x) \) at the given points: \( \frac{5}{2}, \pi, -\pi, \frac{5}{4} \).

- **For \( f\left(\frac{5}{2}\right) \)**:
\[ f\left(\frac{5}{2}\right) = \cos\left(9 \cdot \frac{5}{2}\right) + \cos\left(10 \cdot \frac{5}{2}\right) = \cos\left(\frac{45}{2}\right) + \cos(25) \]
Since \(\frac{45}{2} = 22.5\), we can find the cosine values.

- **For \( f(\pi) \)**:
\[ f(\pi) = \cos(9\pi) + \cos(10\pi) = -1 + 1 = 0 \]

- **For \( f(-\pi) \)**:
\[ f(-\pi) = \cos(-9\pi) + \cos(-10\pi) = -1 + 1 = 0 \]

- **For \( f\left(\frac{5}{4}\right) \)**:
\[ f\left(\frac{5}{4}\right) = \cos\left(9 \cdot \frac{5}{4}\right) + \cos\left(10 \cdot \frac{5}{4}\right) = \cos\left(\frac{45}{4}\right) + \cos\left(\frac{50}{4}\right) \]

5. **Conclusion**:
After evaluating all the points, we can compare the results to the options provided in the question to determine which is correct.