Let `f(x) = cos(x) + cos(sqrt(3)x) AA x in R`. The number of values of x for which f(x) is maximum is

Options

A. 1
B. 2
C. 5
D. infinite

Answer

To solve the problem of finding the number of values of \( x \) for which the function \( f(x) = \cos(x) + \cos(\sqrt{3}x) \) is maximum, we can follow these steps:

### Step 1: Understand the Function
The function \( f(x) \) is the sum of two cosine functions:
- \( \cos(x) \) has a period of \( 2\pi \).
- \( \cos(\sqrt{3}x) \) has a period of \( \frac{2\pi}{\sqrt{3}} \).

### Step 2: Determine the Maximum Value of Each Component
The maximum value of both \( \cos(x) \) and \( \cos(\sqrt{3}x) \) is \( 1 \). Therefore, the maximum value of \( f(x) \) can be calculated as:
\[ f(x)_{\text{max}} = \cos(x) + \cos(\sqrt{3}x) = 1 + 1 = 2. \]

### Step 3: Find Conditions for Maximum
For \( f(x) \) to achieve its maximum value of \( 2 \), both \( \cos(x) \) and \( \cos(\sqrt{3}x) \) must equal \( 1 \) simultaneously. This occurs when:
\[ \cos(x) = 1 \quad \text{and} \quad \cos(\sqrt{3}x) = 1. \]

### Step 4: Solve for \( x \)
The conditions for \( \cos(x) = 1 \) and \( \cos(\sqrt{3}x) = 1 \) give us:
1. \( x = 2n\pi \) for \( n \in \mathbb{Z} \) (from \( \cos(x) = 1 \)).
2. \( \sqrt{3}x = 2m\pi \) for \( m \in \mathbb{Z} \) (from \( \cos(\sqrt{3}x) = 1 \)), which simplifies to \( x = \frac{2m\pi}{\sqrt{3}} \).

### Step 5: Find Common Solutions
To find common solutions, we set:
\[ 2n\pi = \frac{2m\pi}{\sqrt{3}}. \]
Dividing both sides by \( 2\pi \) gives:
\[ n = \frac{m}{\sqrt{3}}. \]
This implies that \( m \) must be a multiple of \( \sqrt{3} \), which is not possible since \( m \) and \( n \) are integers.

### Step 6: Conclusion
Since there are no integer solutions \( n \) and \( m \) that satisfy the equation simultaneously, the only solution occurs when both functions are at their maximum value at \( x = 0 \).

Thus, the number of values of \( x \) for which \( f(x) \) is maximum is:
\[ \boxed{1}. \]