If `f(x-y), f(x) f(y), and f(x+y)` are in A.P. for all x, y, and `f(0) ne 0,` then

Options

A. `f(4)=f(-4)`
B. `f(2)+f(-2)=0`
C. `f'(4)+f'(-4)=0`
D. `f'(2)=f'(-2)`

Explanation

To solve the problem, we need to show that if \( f(x-y), f(x)f(y), \) and \( f(x+y) \) are in arithmetic progression (A.P.) for all \( x \) and \( y \), and \( f(0) \neq 0 \), then we can derive certain properties of the function \( f \).

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
Since \( f(x-y), f(x)f(y), f(x+y) \) are in A.P., we can write the condition for A.P.:
\[
2f(x)f(y) = f(x-y) + f(x+y)
...