A function f :`R to R` satisfies `sin x cos y [f (2x +2y) - f(2x - 2y)] =cos x sin y [f(2x +2y) + f (2x - 2y)]` If `f'(0) = 1/2`, then f (x) =

Options

A. f"(x)
B. `- F"(x)`
C. ` 4f"(x)`
D. `- 4f ''(x)`

Answer

To solve the given problem step by step, we start with the functional equation provided:

**Given:**
\[ \sin x \cos y [f(2x + 2y) - f(2x - 2y)] = \cos x \sin y [f(2x + 2y) + f(2x - 2y)] \]

**Step 1: Rearranging the equation**
We can rearrange the equation to isolate terms involving \( f(2x + 2y) \) and \( f(2x - 2y) \).

\[ \sin x \cos y f(2x + 2y) - \sin x \cos y f(2x - 2y) = \cos x \sin y f(2x + 2y) + \cos x \sin y f(2x - 2y) \]

**Step 2: Collecting like terms**
We can collect the terms involving \( f(2x + 2y) \) on one side and those involving \( f(2x - 2y) \) on the other side:

\[ \sin x \cos y f(2x + 2y) - \cos x \sin y f(2x + 2y) = \sin x \cos y f(2x - 2y) + \cos x \sin y f(2x - 2y) \]

**Step 3: Factoring out common terms**
Factoring out \( f(2x + 2y) \) and \( f(2x - 2y) \):

\[ f(2x + 2y) (\sin x \cos y - \cos x \sin y) = f(2x - 2y) (\sin x \cos y + \cos x \sin y) \]

Using the sine addition and subtraction formulas, we can rewrite this as:

\[ f(2x + 2y) \sin(x - y) = f(2x - 2y) \sin(x + y) \]

**Step 4: Setting up a ratio**
We can express the relationship as:

\[ \frac{f(2x + 2y)}{f(2x - 2y)} = \frac{\sin(x + y)}{\sin(x - y)} \]

**Step 5: Substituting \( x + y = \alpha \)**
Let \( \alpha = x + y \). Then \( x - y = \alpha - 2y \):

\[ \frac{f(2\alpha)}{f(2(\alpha - 2y))} = \frac{\sin \alpha}{\sin(\alpha - 2y)} \]

**Step 6: Finding a general form for \( f(x) \)**
Assuming \( f(x) \) is of the form \( f(x) = r \sin\left(\frac{x}{2}\right) \), we can differentiate to find \( f'(0) \).

**Step 7: Using the condition \( f'(0) = \frac{1}{2} \)**
Differentiating \( f(x) \):

\[ f'(x) = r \cdot \frac{1}{2} \cos\left(\frac{x}{2}\right) \]

Evaluating at \( x = 0 \):

\[ f'(0) = r \cdot \frac{1}{2} \cdot 1 = \frac{r}{2} \]

Setting this equal to \( \frac{1}{2} \):

\[ \frac{r}{2} = \frac{1}{2} \implies r = 1 \]

**Step 8: Final form of \( f(x) \)**
Thus, we have:

\[ f(x) = \sin\left(\frac{x}{2}\right) \]

### Final Answer:
\[ f(x) = \sin\left(\frac{x}{2}\right) \]