Let f(x) =sin `({x})/(a) + cos ({x})/(a)` where `a gt 0` and {.} denotes the fractional part function . Then find the set of values of 'a' for which f can attain its maximum values.

Answer

To solve the problem, we need to analyze the function given by:

\[ f(x) = \sin\left(\frac{\{x\}}{a}\right) + \cos\left(\frac{\{x\}}{a}\right) \]

where \(\{x\}\) denotes the fractional part of \(x\), and \(a > 0\).

### Step 1: Understanding the Fractional Part Function
The fractional part function \(\{x\}\) gives us the value of \(x\) between 0 and 1. For example:
- \(\{1.8\} = 0.8\)
- \(\{-1.3\} = 0.7\)

Thus, \(\{x\}\) will always lie in the interval \([0, 1)\).

### Step 2: Rewrite the Function
Let \(\theta = \frac{\{x\}}{a}\). Then we can rewrite the function as:

\[ f(x) = \sin(\theta) + \cos(\theta) \]

### Step 3: Use Trigonometric Identity
We can express \(\sin(\theta) + \cos(\theta)\) using the identity:

\[ \sin(\theta) + \cos(\theta) = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \]

This means:

\[ f(x) = \sqrt{2} \sin\left(\frac{\{x\}}{a} + \frac{\pi}{4}\right) \]

### Step 4: Finding Maximum Value
The maximum value of \(\sin\) function is 1. Therefore, for \(f(x)\) to attain its maximum value, we need:

\[ \frac{\{x\}}{a} + \frac{\pi}{4} = \frac{\pi}{2} + k\pi \quad (k \in \mathbb{Z}) \]

This simplifies to:

\[ \frac{\{x\}}{a} = \frac{\pi}{4} + k\pi \]

### Step 5: Solve for 'a'
From the equation above, we can express \(a\) in terms of \(\{x\}\):

\[ a = \frac{\{x\}}{\frac{\pi}{4} + k\pi} \]

### Step 6: Determine the Range of 'a'
Since \(\{x\}\) lies in the interval \([0, 1)\), we analyze the values of \(a\):

1. For \(k = 0\):
\[ a = \frac{\{x\}}{\frac{\pi}{4}} \]
This gives a maximum value of \(a\) as:
\[ a < \frac{1}{\frac{\pi}{4}} = \frac{4}{\pi} \]

2. For \(k = -1\):
\[ a = \frac{\{x\}}{\frac{\pi}{4} - \pi} = \frac{\{x\}}{-\frac{3\pi}{4}} \]
This is not valid since \(a\) must be positive.

3. For \(k = 1\):
\[ a = \frac{\{x\}}{\frac{\pi}{4} + \pi} = \frac{\{x\}}{\frac{5\pi}{4}} \]
This gives a maximum value of \(a\) as:
\[ a < \frac{1}{\frac{5\pi}{4}} = \frac{4}{5\pi} \]

### Conclusion
The set of values for \(a\) such that \(f(x)\) can attain its maximum value is:

\[ 0 < a < \frac{4}{\pi} \]