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

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.