The maximum value of
`sin(x+pi/5)+cos(x+pi/5)`,where `x epsilon (0,pi/2)`, is attained at:

Options

A. `(pi)/(20)`
B. `(pi)/(15)`
C. `(pi)/(10)`
D. `(pi)/(2)`

Answer

To find the maximum value of the expression \( \sin(x + \frac{\pi}{5}) + \cos(x + \frac{\pi}{5}) \) for \( x \in (0, \frac{\pi}{2}) \), we can follow these steps:

### Step 1: Rewrite the expression
We can rewrite the expression using the identity for the sum of sine and cosine:
\[ \sin(a) + \cos(a) = \sqrt{2} \sin\left(a + \frac{\pi}{4}\right) \]
where \( a = x + \frac{\pi}{5} \).

### Step 2: Apply the identity
Thus, we have:
\[ \sin(x + \frac{\pi}{5}) + \cos(x + \frac{\pi}{5}) = \sqrt{2} \sin\left(x + \frac{\pi}{5} + \frac{\pi}{4}\right) \]
This simplifies to:
\[ \sqrt{2} \sin\left(x + \frac{9\pi}{20}\right) \]

### Step 3: Find the maximum value of the sine function
The sine function achieves its maximum value of 1. Therefore, the maximum value of our expression is:
\[ \sqrt{2} \cdot 1 = \sqrt{2} \]

### Step 4: Determine when the maximum occurs
To find when this maximum occurs, we need:
\[ x + \frac{9\pi}{20} = \frac{\pi}{2} \]
Solving for \( x \):
\[ x = \frac{\pi}{2} - \frac{9\pi}{20} \]
Finding a common denominator (which is 20):
\[ x = \frac{10\pi}{20} - \frac{9\pi}{20} = \frac{\pi}{20} \]

### Conclusion
Thus, the maximum value of \( \sin(x + \frac{\pi}{5}) + \cos(x + \frac{\pi}{5}) \) is attained at:
\[ x = \frac{\pi}{20} \]

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