Determine the value(s) of $$ k $$ for which $$ f(x)=k $$ will have two unequal positive real roots.

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To determine the values of $$ k $$ for which the equation $$ f(x) = k $$ has two unequal positive real roots, we need to analyze the function $$ f(x) $$ and its behavior.

1. **Identify the function**: We need to know the specific form of $$ f(x) $$. If $$ f(x) $$ is a polynomial or another type of function, we can analyze its properties (like its derivative, critical points, and concavity).

2. **Conditions for two unequal positive real roots**:
   - The function $$ f(x) $$ must intersect the line $$ y = k $$ at two points.
   - The value of $$ k $$ must be such that the intersection points are both positive and unequal.

3. **Finding critical points**: We will find the derivative $$ f'(x) $$ and set it to zero to find critical points. This will help us determine the local maxima and minima of $$ f(x) $$.

4. **Analyzing the function**: We will evaluate the function at the critical points and check the values of $$ f(x) $$ to find the range of $$ k $$ that allows for two intersections.

5. **Determine the range of $$ k $$**: We will find the maximum and minimum values of $$ f(x) $$ at the critical points and determine the range of $$ k $$ that allows for two unequal positive roots.

Please provide the specific form of $$ f(x) $$ so that we can proceed with the calculations.
To find the values of $$ k $$ for which $$ f(x) = k $$ has two unequal positive real roots, analyze the function $$ f(x) $$ by finding its critical points and determining the range of $$ k $$ where the equation intersects the line $$ y = k $$ at two distinct positive points.

Determine the value(s) of \( k \) for which \( f(x)=k \) will have two unequal positive real roots.

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