If `f((xy)/(2)) = (f(x).f(y))/(2), x, y in R, f(1) = f'(1)."Then", (f(3))/(f'(3))`is.........

Answer

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

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

for all \(x, y \in \mathbb{R}\), and we know that \(f(1) = f'(1)\). We need to find the value of \(\frac{f(3)}{f'(3)}\).

### Step 1: Assume a Form for \(f(x)\)

Let's assume that \(f(x) = x^a\) for some constant \(a\). This is a common approach for functional equations of this type.

### Step 2: Substitute into the Functional Equation

Substituting \(f(x) = x^a\) into the functional equation gives:

\[ f\left(\frac{xy}{2}\right) = \left(\frac{xy}{2}\right)^a = \frac{(xy)^a}{2^a} \]

On the right side, we have:

\[ \frac{f(x)f(y)}{2} = \frac{x^a y^a}{2} = \frac{(xy)^a}{2} \]

### Step 3: Equate Both Sides

Setting both sides equal, we have:

\[ \frac{(xy)^a}{2^a} = \frac{(xy)^a}{2} \]

For this to hold for all \(x\) and \(y\), we must have:

\[ 2^a = 2 \implies a = 1 \]

### Step 4: Determine \(f(x)\)

Thus, we find that:

\[ f(x) = x^1 = x \]

### Step 5: Compute \(f(1)\) and \(f'(1)\)

Now we compute \(f(1)\) and \(f'(1)\):

\[ f(1) = 1 \]

To find \(f'(x)\), we differentiate \(f(x)\):

\[ f'(x) = 1 \]

Thus,

\[ f'(1) = 1 \]

Since \(f(1) = f'(1)\), this condition is satisfied.

### Step 6: Compute \(f(3)\) and \(f'(3)\)

Now, we compute \(f(3)\) and \(f'(3)\):

\[ f(3) = 3 \]
\[ f'(3) = 1 \]

### Step 7: Find \(\frac{f(3)}{f'(3)}\)

Finally, we find:

\[ \frac{f(3)}{f'(3)} = \frac{3}{1} = 3 \]

Thus, the answer is:

\[ \boxed{3} \]