Let `f(x) = sin(pi[x]) + sin([pi^(2)]x) + cos ([-pi^(2)]x/3) AA x in R`, then `f(pi//4)` is equal to

Options

A. `1//sqrt(2)`
B. `-1//sqrt(2)`
C. `sqrt(2)`
D. `-sqrt(2)`

Answer

To solve the problem, we need to evaluate the function \( f(x) = \sin(\pi [x]) + \sin([\pi^2] x) + \cos\left(-\frac{\pi^2 x}{3}\right) \) at \( x = \frac{\pi}{4} \).

### Step 1: Calculate \( [x] \) for \( x = \frac{\pi}{4} \)

The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \).

\[ x = \frac{\pi}{4} \approx 0.785 \quad \text{(since } \pi \approx 3.14\text{)} \]

Thus,

\[ [\frac{\pi}{4}] = 0 \]

### Step 2: Substitute \( [x] \) into \( f(x) \)

Now substitute \( [x] \) into the function:

\[ f\left(\frac{\pi}{4}\right) = \sin(\pi [\frac{\pi}{4}]) + \sin([\pi^2] \cdot \frac{\pi}{4}) + \cos\left(-\frac{\pi^2 \cdot \frac{\pi}{4}}{3}\right) \]

This simplifies to:

\[ f\left(\frac{\pi}{4}\right) = \sin(\pi \cdot 0) + \sin([\pi^2] \cdot \frac{\pi}{4}) + \cos\left(-\frac{\pi^3}{12}\right) \]

### Step 3: Calculate \( \sin(\pi \cdot 0) \)

\[ \sin(0) = 0 \]

### Step 4: Calculate \( [\pi^2] \)

Next, we need to find \( [\pi^2] \):

\[ \pi^2 \approx 9.87 \quad \Rightarrow \quad [\pi^2] = 9 \]

### Step 5: Substitute \( [\pi^2] \) into the function

Now substitute \( [\pi^2] \):

\[ f\left(\frac{\pi}{4}\right) = 0 + \sin(9 \cdot \frac{\pi}{4}) + \cos\left(-\frac{\pi^3}{12}\right) \]

### Step 6: Calculate \( \sin(9 \cdot \frac{\pi}{4}) \)

\[ 9 \cdot \frac{\pi}{4} = \frac{9\pi}{4} \]

To simplify \( \sin\left(\frac{9\pi}{4}\right) \):

\[ \frac{9\pi}{4} = 2\pi + \frac{\pi}{4} \quad \Rightarrow \quad \sin\left(\frac{9\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 7: Calculate \( \cos\left(-\frac{\pi^3}{12}\right) \)

Since cosine is an even function:

\[ \cos\left(-\frac{\pi^3}{12}\right) = \cos\left(\frac{\pi^3}{12}\right) \]

### Step 8: Combine the results

Now we can combine everything:

\[ f\left(\frac{\pi}{4}\right) = 0 + \frac{1}{\sqrt{2}} + \cos\left(\frac{\pi^3}{12}\right) \]

### Final Result

Thus, the final value of \( f\left(\frac{\pi}{4}\right) \) is:

\[ f\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + \cos\left(\frac{\pi^3}{12}\right) \]