If `f(x+10) + f(x+4)= 0`, there `f(x)` is a periodic function with period

Options

A. 2
B. 4
C. 6
D. 12

Answer

To solve the problem, we need to determine the period of the periodic function \( f(x) \) given the equation:

\[ f(x + 10) + f(x + 4) = 0 \]

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ f(x + 10) + f(x + 4) = 0 \]

2. **Replace \( x \) with \( x - 10 \)**:
\[ f((x - 10) + 10) + f((x - 10) + 4) = 0 \]
This simplifies to:
\[ f(x) + f(x - 6) = 0 \]
(Let’s call this Equation 1)

3. **Now, replace \( x \) with \( x - 4 \)** in the original equation:
\[ f((x - 4) + 10) + f((x - 4) + 4) = 0 \]
This simplifies to:
\[ f(x + 6) + f(x) = 0 \]
(Let’s call this Equation 2)

4. **Now we have two equations**:
- From Equation 1: \( f(x) + f(x - 6) = 0 \)
- From Equation 2: \( f(x + 6) + f(x) = 0 \)

5. **Equate the two equations**:
From Equation 1, we can express \( f(x - 6) \):
\[ f(x - 6) = -f(x) \]
From Equation 2, we can express \( f(x + 6) \):
\[ f(x + 6) = -f(x) \]

6. **Now, set \( f(x + 6) \) equal to \( f(x - 6) \)**:
\[ f(x + 6) = f(x - 6) \]

7. **This implies that**:
\[ f(x) = f(x - 12) \]
This means \( f(x) \) is periodic with period \( 12 \).

### Conclusion:
The period of the function \( f(x) \) is \( 12 \).