If ` f: (0, oo) ->(0, oo)` for which there is a positive real number a such that it satisfies differential equation `f'(a/x)=x/(f(x))` , then

Options

A. `f(x)` can be linear
B. `f(x)` can be a functional of the type `m(x)^(1//n),m epsilonR^(+),n epsilonI^(+)`
C. `f^(')(x)` can be positive
D. `f(x)` can be twice differentiable

Explanation

To solve the problem step by step, we will analyze the given differential equation and explore the properties of the function \( f(x) \).

### Step 1: Understand the Given Information
We are given a function \( f: (0, \infty) \to (0, \infty) \) and a differential equation:

\[ f'\left(\frac{a}{x}\right) = \frac{x}{f(x)} \]

where \( a \) is a positive real number.

### Step 2: Assume a Form for \( f(x) \)
Let's assume that \( f(x) \) is a linear function of the form:

\[ f(x) = ax + b \]

where \( a \) and \( b \) are constants.

### Step 3: Calculate the Derivative
The derivative of \( f(x) \) is:

\[ f'(x) = a \]

Now, substituting \( \frac{a}{x} \) into the derivative:

\[ f'\left(\frac{a}{x}\right) = a \]

### Step 4: Substitute into the Differential Equation
Now substituting into the differential equation:

\[ a = \frac{x}{f(x)} = \frac{x}{ax + b} \]

### Step 5: Cross-Multiply and Rearrange
Cross-multiplying gives us:

\[ a(ax + b) = x \]

Expanding this yields:

\[ a^2x + ab = x \]

### Step 6: Compare Coefficients
Now we compare coefficients of \( x \):

1. From \( a^2x = x \), we get \( a^2 = 1 \) which implies \( a = 1 \) or \( a = -1 \).
2. The constant term gives us \( ab = 0 \).

### Step 7: Determine Values of \( a \) and \( b \)
From \( ab = 0 \):
- If \( a = 1 \), then \( b = 0 \).
- If \( a = -1 \), then \( b = 0 \).

Thus, we have two possible functions:

\[ f(x) = x \quad \text{or} \quad f(x) = -x \]

However, since \( f(x) \) must be positive for \( x > 0 \), we discard \( f(x) = -x \).

### Step 8: Verify Properties of \( f(x) \)
1. **Linear Function**: \( f(x) = x \) is linear.
2. **Twice Differentiable**: The function \( f(x) = x \) is differentiable and its second derivative is zero, confirming it is twice differentiable.
3. **Positive Derivative**: The derivative \( f'(x) = 1 \) is positive.

### Conclusion
The function \( f(x) = x \) satisfies all the conditions given in the problem. Therefore, we conclude:

- \( f(x) \) can be linear.
- \( f(x) \) can be twice differentiable.
- \( f'(x) \) can be positive.