Let f be a real valued function such that for any real x `f(15+x)=f(15-x)and f(30+x)=-f(30-x)`
Then which of the following statements is true ?

Options

A. f is odd and periodic
B. f is odd but not periodic
C. f is even and periodic
D. f is even nut not periodic

Answer

To solve the problem, we need to analyze the given properties of the function \( f \):

1. **Given Conditions**:
- \( f(15 + x) = f(15 - x) \)
- \( f(30 + x) = -f(30 - x) \)

2. **Step 1: Analyzing the first condition**:
The first condition \( f(15 + x) = f(15 - x) \) indicates that the function is symmetric about \( x = 15 \). This means that for any \( x \), the value of the function at \( 15 + x \) is the same as at \( 15 - x \).

**Hint**: This symmetry suggests that if you were to plot the function, it would be mirrored around the vertical line \( x = 15 \).

3. **Step 2: Analyzing the second condition**:
The second condition \( f(30 + x) = -f(30 - x) \) indicates that the function is antisymmetric about \( x = 30 \). This means that for any \( x \), the value of the function at \( 30 + x \) is the negative of the value at \( 30 - x \).

**Hint**: This antisymmetry suggests that the function has rotational symmetry around the point \( (30, 0) \).

4. **Step 3: Finding specific values**:
Let's substitute \( x = 0 \) in both conditions:
- From the first condition: \( f(15) = f(15) \) (which is trivially true).
- From the second condition: \( f(30) = -f(30) \) implies \( f(30) = 0 \).

**Hint**: Finding \( f(30) = 0 \) is crucial as it gives us a specific point on the function.

5. **Step 4: Exploring further values**:
Now, let's explore what happens when we substitute \( x = 15 \) in the second condition:
- \( f(30 + 15) = -f(30 - 15) \) gives \( f(45) = -f(15) \).

**Hint**: This relationship shows how values of the function relate to each other.

6. **Step 5: Establishing periodicity**:
Next, let's examine what happens when we replace \( x \) with \( x + 30 \) in the second condition:
- \( f(30 + (x + 30)) = -f(30 - (x + 30)) \) simplifies to \( f(60 + x) = -f(-x) \).

**Hint**: This indicates a periodic behavior, suggesting that the function may repeat every 60 units.

7. **Step 6: Establishing oddness**:
From the second condition, we also note that \( f(30 + x) = -f(30 - x) \) implies that if we replace \( x \) with \( -x \), we get \( f(30 - x) = -f(30 + x) \), which shows that \( f(-x) = -f(x) \).

**Hint**: This confirms that the function is odd, meaning \( f(-x) = -f(x) \) for all \( x \).

8. **Conclusion**:
From the analysis, we conclude that the function \( f \) is both periodic with a period of 60 and odd. Therefore, the correct statement is that the function is periodic and odd.

**Final Answer**: The function \( f \) is periodic with period 60 and odd.