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

Options

A. `(pi)/(12)`
B. `(pi)/(6)`
C. `(pi)/(3)`
D. `(pi)/(20)`

Answer

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

### Step 1: Define the Function
Let \( \theta = x + \frac{\pi}{5} \). Then, we can rewrite the function as:
\[ f(\theta) = \sin(\theta) + \cos(\theta) \]

### Step 2: Differentiate the Function
To find the maximum value, we differentiate \( f(\theta) \) with respect to \( \theta \):
\[ f'(\theta) = \cos(\theta) - \sin(\theta) \]

### Step 3: Set the Derivative to Zero
To find the critical points, we set the derivative equal to zero:
\[ \cos(\theta) - \sin(\theta) = 0 \]
This implies:
\[ \cos(\theta) = \sin(\theta) \]

### Step 4: Solve for \( \theta \)
From \( \cos(\theta) = \sin(\theta) \), we can conclude that:
\[ \tan(\theta) = 1 \]
This occurs at:
\[ \theta = \frac{\pi}{4} \]

### Step 5: Relate \( \theta \) Back to \( x \)
Recall that \( \theta = x + \frac{\pi}{5} \). Therefore, we set:
\[ x + \frac{\pi}{5} = \frac{\pi}{4} \]

### Step 6: Solve for \( x \)
Now, we solve for \( x \):
\[ x = \frac{\pi}{4} - \frac{\pi}{5} \]
To simplify this, we find a common denominator:
\[ x = \frac{5\pi}{20} - \frac{4\pi}{20} = \frac{\pi}{20} \]

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