Let f be a function defined from `R^(+)rarrR^(+).` If `(f(xy))^(2)=x(f(y))^(2)` for all positive numbers x and y, If `f(2)=6,` find `f(50)`=?

Options

A. 20
B. 30
C. 5
D. 40

Explanation

To solve the problem step by step, we will use the given functional equation and the known value of the function at a specific point.

### Step 1: Write down the functional equation
We are given the functional equation:
\[ (f(xy))^2 = x(f(y))^2 \]
for all positive numbers \(x\) and \(y\).

### Step 2: Substitute specific values into the functional equation
We know that \(f(2) = 6\). Let's substitute \(x = 25\) and \(y = 2\) into the functional equation.

\[ f(25 \cdot 2)^2 = 25(f(2))^2 \]

### Step 3: Simplify the left side
Calculating the left side:
\[ f(50)^2 = 25(f(2))^2 \]

### Step 4: Substitute the known value of \(f(2)\)
Now substitute \(f(2) = 6\) into the equation:
\[ f(50)^2 = 25 \cdot (6)^2 \]

### Step 5: Calculate \( (6)^2 \)
Calculating \( (6)^2 \):
\[ (6)^2 = 36 \]

### Step 6: Substitute back into the equation
Now substitute back:
\[ f(50)^2 = 25 \cdot 36 \]

### Step 7: Calculate \( 25 \cdot 36 \)
Calculating \( 25 \cdot 36 \):
\[ 25 \cdot 36 = 900 \]

### Step 8: Take the square root
Now take the square root of both sides to find \(f(50)\):
\[ f(50) = \sqrt{900} \]

### Step 9: Calculate the square root
Calculating the square root:
\[ f(50) = 30 \]

### Final Answer
Thus, the value of \(f(50)\) is:
\[ \boxed{30} \]