If f(x) = sinx + cos ax is a periodic function then a cannot be

Options

A. `pi`
B. `sqrt2`
C. 2
D. `1/2`

Answer

To determine the values of \( a \) for which the function \( f(x) = \sin x + \cos(ax) \) is a periodic function, we will follow these steps:

### Step 1: Understand the periodicity of the components
The function \( f(x) \) is the sum of two functions: \( \sin x \) and \( \cos(ax) \). The periodicity of a function is defined as the smallest positive \( T \) such that \( f(x + T) = f(x) \) for all \( x \).

### Step 2: Find the period of \( \sin x \)
The period of \( \sin x \) is \( 2\pi \). This means:
\[ T_1 = 2\pi \]

### Step 3: Find the period of \( \cos(ax) \)
The period of \( \cos(ax) \) is given by:
\[ T_2 = \frac{2\pi}{|a|} \]

### Step 4: Determine the overall period of \( f(x) \)
For \( f(x) \) to be periodic, the periods \( T_1 \) and \( T_2 \) must be commensurable, meaning there exists a least common multiple (LCM) of the two periods. The overall period \( T \) can be expressed as:
\[ T = \text{lcm}(T_1, T_2) = \text{lcm}(2\pi, \frac{2\pi}{|a|}) \]

### Step 5: Analyze the condition for periodicity
For the function \( f(x) \) to be periodic, \( \frac{2\pi}{|a|} \) must be a rational number. This implies that \( |a| \) must be a rational number.

### Step 6: Identify the values of \( a \)
If \( a \) is an irrational number, then \( \frac{2\pi}{|a|} \) will also be irrational, and thus \( f(x) \) will not be periodic. Therefore, \( a \) cannot be irrational.

### Conclusion
The values of \( a \) that make \( f(x) \) a periodic function must be rational. Thus, the answer to the question "a cannot be" is that \( a \) cannot be an irrational number.