Let `f(x)=1/2[f(xy)+f(x/y)] " for " x,y in R^(+)` such that f(1)=0,f'(1)=2.`
f(x)-f(y) is equal to

Options

A. `f(y/x)`
B. `f(x/y)`
C. f(2x)
D. f(2y)

Answer

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

\[ f(x) = \frac{1}{2} [f(xy) + f(x/y)] \]

for \( x, y \in \mathbb{R}^{+} \), with the conditions \( f(1) = 0 \) and \( f'(1) = 2 \).

### Step 1: Analyze the functional equation

We can rewrite the functional equation as:

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

This suggests a logarithmic form for \( f(x) \). Let's assume:

\[ f(x) = k \log x \]

for some constant \( k \).

### Step 2: Substitute the assumed form into the functional equation

Substituting \( f(x) = k \log x \) into the functional equation gives:

\[ 2(k \log x) = k \log(xy) + k \log\left(\frac{x}{y}\right) \]

Using properties of logarithms, we have:

\[ 2k \log x = k (\log x + \log y) + k (\log x - \log y) \]

### Step 3: Simplify the equation

Now, simplifying the right side:

\[ k (\log x + \log y + \log x - \log y) = k (2 \log x) \]

So we have:

\[ 2k \log x = 2k \log x \]

This confirms that our assumption \( f(x) = k \log x \) satisfies the functional equation.

### Step 4: Use the conditions to find \( k \)

Now we need to determine the value of \( k \) using the conditions provided.

1. From \( f(1) = 0 \):

\[ f(1) = k \log(1) = k \cdot 0 = 0 \]

This condition is satisfied for any \( k \).

2. From \( f'(1) = 2 \):

We first find the derivative of \( f(x) \):

\[ f'(x) = k \cdot \frac{1}{x} \]

Evaluating this at \( x = 1 \):

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

Setting this equal to the condition:

\[ k = 2 \]

### Step 5: Write the final form of \( f(x) \)

Thus, we have:

\[ f(x) = 2 \log x \]

### Step 6: Find \( f(x) - f(y) \)

Now we calculate \( f(x) - f(y) \):

\[ f(x) - f(y) = (2 \log x) - (2 \log y) \]

Factoring out the common term:

\[ f(x) - f(y) = 2 (\log x - \log y) \]

Using the property of logarithms:

\[ \log x - \log y = \log\left(\frac{x}{y}\right) \]

Thus, we can write:

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

### Final Result

So, the final answer is:

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

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