Let [x] = the greatest integer less than or equal to x and let `f (x) = sin x + cosx.` Then the most general solution of `f(x)=[f(pi/10)]` is

Answer

To solve the problem, we need to find the most general solution of the equation \( f(x) = [f(\pi/10)] \), where \( f(x) = \sin x + \cos x \) and \([x]\) denotes the greatest integer less than or equal to \(x\).

### Step-by-Step Solution:

1. **Calculate \( f(\pi/10) \)**:
\[ f\left(\frac{\pi}{10}\right) = \sin\left(\frac{\pi}{10}\right) + \cos\left(\frac{\pi}{10}\right) \]

2. **Estimate the value of \( f(\pi/10) \)**:
We know that both \(\sin\) and \(\cos\) functions are positive in the first quadrant. Therefore, we can estimate:
\[ \sin\left(\frac{\pi}{10}\right) \approx 0.309 \quad \text{and} \quad \cos\left(\frac{\pi}{10}\right) \approx 0.951 \]
Adding these gives:
\[ f\left(\frac{\pi}{10}\right) \approx 0.309 + 0.951 = 1.26 \]

3. **Find the greatest integer function**:
Since \( f\left(\frac{\pi}{10}\right) \approx 1.26 \), we have:
\[ [f(\pi/10)] = 1 \]

4. **Set up the equation**:
Now we need to solve:
\[ f(x) = 1 \quad \Rightarrow \quad \sin x + \cos x = 1 \]

5. **Rearranging the equation**:
We can rewrite the equation as:
\[ \sin x + \cos x - 1 = 0 \]

6. **Using the identity**:
We can express \(\sin x + \cos x\) in terms of a single cosine function:
\[ \sin x + \cos x = \sqrt{2} \left(\sin x \cdot \frac{1}{\sqrt{2}} + \cos x \cdot \frac{1}{\sqrt{2}}\right) = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]
Thus, we have:
\[ \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) = 1 \]

7. **Solving for \(\sin\)**:
Dividing both sides by \(\sqrt{2}\):
\[ \sin\left(x + \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} = \sin\left(\frac{\pi}{4}\right) \]

8. **Finding the general solution**:
The general solution for \(\sin A = \sin B\) is:
\[ A = B + 2n\pi \quad \text{or} \quad A = \pi - B + 2n\pi \]
Applying this:
\[ x + \frac{\pi}{4} = \frac{\pi}{4} + 2n\pi \quad \text{or} \quad x + \frac{\pi}{4} = \pi - \frac{\pi}{4} + 2n\pi \]

9. **Solving the equations**:
- From \(x + \frac{\pi}{4} = \frac{\pi}{4} + 2n\pi\):
\[ x = 2n\pi \]
- From \(x + \frac{\pi}{4} = \frac{3\pi}{4} + 2n\pi\):
\[ x = 2n\pi + \frac{\pi}{2} \]

10. **Final general solutions**:
Therefore, the most general solutions are:
\[ x = 2n\pi \quad \text{and} \quad x = 2n\pi + \frac{\pi}{2} \]