1. Let's clarify the concepts of local minimum and absolute maximum. 2. A local minimum at a point means the function's value there is lower than all nearby points. 3. An absolute maximum means the function's value at that point is greater than or equal to all other points in the domain. 4. It is possible for a function to have a local minimum and an absolute maximum with the same value if the function is constant or has flat regions. 5. For example, if the function equals 5 at a local minimum and nowhere else exceeds 5, then 5 is also the absolute maximum. 6. So, having a local minimum value of 5 and an absolute maximum value of 5 is consistent if the function does not go above 5 anywhere. 7. This situation often occurs in functions that are constant or have plateaus. Final answer: It makes sense if the function's value at the local minimum is 5 and no other value exceeds 5, making 5 both a local minimum and the absolute maximum.