If `f(3x+2)+f(3x+29)=0 AAx in R ,` then the period of `f(x)` is

Options

A. 7
B. 8
C. 10
D. none of these

Explanation

To find the period of the function \( f(x) \) given the equation \( f(3x + 2) + f(3x + 29) = 0 \) for all \( x \in \mathbb{R} \), we can follow these steps:

### Step 1: Substitute \( x \) with \( x + 9 \)
We start by substituting \( x \) with \( x + 9 \) in the original equation:
\[ f(3(x + 9) + 2) + f(3(x + 9) + 29) = 0 \]
This simplifies to:
\[ f(3x + 27 + 2) + f(3x + 27 + 29) = 0 \]
Which further simplifies to:
\[ f(3x + 29) + f(3x + 56) = 0 \]

### Step 2: Equate the two equations
Now we have two equations:
1. \( f(3x + 2) + f(3x + 29) = 0 \) (Equation 1)
2. \( f(3x + 29) + f(3x + 56) = 0 \) (Equation 2)

From Equation 1, we can express \( f(3x + 2) \):
\[ f(3x + 2) = -f(3x + 29) \]
From Equation 2, we can express \( f(3x + 56) \):
\[ f(3x + 56) = -f(3x + 29) \]

### Step 3: Relate \( f(3x + 2) \) and \( f(3x + 56) \)
Now, we can substitute \( f(3x + 29) \) from Equation 1 into the expression from Equation 2:
\[ f(3x + 2) = f(3x + 56) \]

### Step 4: Set \( y = 3x + 2 \)
Let \( y = 3x + 2 \). Then we can rewrite the equation as:
\[ f(y) = f(y + 54) \]
This implies that the function \( f(x) \) is periodic with a period of \( 54 \).

### Conclusion
Thus, the period of the function \( f(x) \) is \( 54 \).