Let the smallest positive value of x for which the function ` f(x)=sin""(x)/(3)+sin""(x)/(11), ( x in R )` achieves its maximum value be ` x_(0)`. Express `x_(0)` in degree i.e. ` x_(0)=alpha^(0)`. Then , the sum of the digits in ` alpha ` is

Options

A. 15
B. 17
C. 16
D. 18

Answer

To solve the problem, we need to find the smallest positive value of \( x \) for which the function

\[ f(x) = \frac{\sin x}{3} + \frac{\sin x}{11} \]

achieves its maximum value. The maximum value of \( \sin x \) is 1, so we need to determine when both \( \frac{\sin x}{3} \) and \( \frac{\sin x}{11} \) are at their maximum simultaneously.

### Step 1: Determine when \( \sin \frac{x}{3} = 1 \)

The sine function achieves its maximum value of 1 at:

\[ \frac{x}{3} = 2m\pi + \frac{\pi}{2} \quad \text{for } m \in \mathbb{Z} \]

Multiplying through by 3 gives:

\[ x = 6m\pi + \frac{3\pi}{2} \]

### Step 2: Determine when \( \sin \frac{x}{11} = 1 \)

Similarly, for \( \sin \frac{x}{11} = 1 \):

\[ \frac{x}{11} = 2n\pi + \frac{\pi}{2} \quad \text{for } n \in \mathbb{Z} \]

Multiplying through by 11 gives:

\[ x = 22n\pi + \frac{11\pi}{2} \]

### Step 3: Set the two expressions for \( x \) equal to each other

We now have two expressions for \( x \):

1. \( x = 6m\pi + \frac{3\pi}{2} \)
2. \( x = 22n\pi + \frac{11\pi}{2} \)

Setting them equal:

\[ 6m\pi + \frac{3\pi}{2} = 22n\pi + \frac{11\pi}{2} \]

### Step 4: Simplify the equation

Rearranging gives:

\[ 6m\pi - 22n\pi = \frac{11\pi}{2} - \frac{3\pi}{2} \]

This simplifies to:

\[ 6m - 22n = 4 \]

### Step 5: Solve for \( m \) and \( n \)

Rearranging gives:

\[ 3m - 11n = 2 \]

We need integer solutions for \( m \) and \( n \). Testing small values:

- Let \( n = 0 \): \( 3m = 2 \) (not an integer)
- Let \( n = 1 \): \( 3m - 11 = 2 \) → \( 3m = 13 \) (not an integer)
- Let \( n = 2 \): \( 3m - 22 = 2 \) → \( 3m = 24 \) → \( m = 8 \)

Thus, \( m = 8 \) and \( n = 2 \) is a valid solution.

### Step 6: Find the smallest positive \( x \)

Substituting \( m = 8 \) into the equation for \( x \):

\[ x = 6(8)\pi + \frac{3\pi}{2} = 48\pi + \frac{3\pi}{2} = 48\pi + 1.5\pi = 49.5\pi \]

Substituting \( n = 2 \) into the equation for \( x \):

\[ x = 22(2)\pi + \frac{11\pi}{2} = 44\pi + 5.5\pi = 49.5\pi \]

Both methods give the same result.

### Step 7: Convert \( x \) to degrees

To convert \( x = 49.5\pi \) radians to degrees:

\[ x = 49.5 \times \frac{180}{\pi} = 49.5 \times 180 \div \pi \approx 2835 \text{ degrees} \]

### Step 8: Find \( \alpha \)

Here, \( \alpha = 2835 \).

### Step 9: Calculate the sum of the digits in \( \alpha \)

The digits of \( 2835 \) are \( 2, 8, 3, 5 \).

Calculating the sum:

\[ 2 + 8 + 3 + 5 = 18 \]

### Final Answer

The sum of the digits in \( \alpha \) is \( \boxed{18} \).