Let f(x) `= sin a x+ cos bx` be a periodic function, then

Options

A. `a = (3pi)/(2), b = pi`
B. `a = sqrt(3), b = 5sqrt(3)`
C. `a = 3sqrt(2), b = 2sqrt(3)`
D. `a, b in R`

Answer

To determine the values of \( a \) and \( b \) such that the function \( f(x) = \sin(ax) + \cos(bx) \) is periodic, we need to analyze the periods of the individual components of the function.

### Step 1: Identify the periods of the sine and cosine functions
The period of the sine function \( \sin(ax) \) is given by:
\[ T_1 = \frac{2\pi}{|a|} \]
The period of the cosine function \( \cos(bx) \) is given by:
\[ T_2 = \frac{2\pi}{|b|} \]

### Step 2: Find the least common multiple (LCM) of the periods
For the function \( f(x) \) to be periodic, the periods \( T_1 \) and \( T_2 \) must be commensurable, meaning there exists a common period \( T \) such that:
\[ T = nT_1 = mT_2 \]
for some integers \( n \) and \( m \).

### Step 3: Set up the equation for commensurability
This leads us to the equation:
\[ n \cdot \frac{2\pi}{|a|} = m \cdot \frac{2\pi}{|b|} \]
Simplifying this, we get:
\[ \frac{n}{|a|} = \frac{m}{|b|} \]
or
\[ n|b| = m|a| \]

### Step 4: Conclusion about the relationship between \( a \) and \( b \)
From the equation \( n|b| = m|a| \), we can conclude that for \( f(x) \) to be periodic, the ratios of \( a \) and \( b \) must be rational. This means:
\[ \frac{a}{b} = \frac{m}{n} \]
for some integers \( m \) and \( n \).

### Final Result
Thus, the values of \( a \) and \( b \) must be such that their ratio is a rational number for the function \( f(x) = \sin(ax) + \cos(bx) \) to be periodic.

---