what is the global extrema if the local maximum is (1,5) and the local minimum is (3,1)

To find the global extrema (global maximum and global minimum) of a function, we need to compare the values at the critical points, including any local maxima and minima, and sometimes endpoints if given a closed interval. However, since you've only provided local extrema and no information about the behavior of the function at infinity or any interval, we can still discuss global extrema under certain assumptions.

Given the local maximum at \( (1, 5) \) and the local minimum at \( (3, 1) \), here's what we can infer:

1. **Local Maximum at (1, 5)**: The function has a peak at \( x = 1 \), where the value of the function is \( 5 \).
2. **Local Minimum at (3, 1)**: The function reaches a trough at \( x = 3 \), where the value of the function is \( 1 \).

### Global Extrema:
- **Global Maximum**: If these are the only critical points, the global maximum would be at \( (1, 5) \), meaning the global maximum value is \( 5 \).
- **Global Minimum**: Similarly, the global minimum would be at \( (3, 1) \), meaning the global minimum value is \( 1 \).

Without additional information (like boundary conditions or the behavior of the function at infinity), the local extrema could be considered the global extrema for now.

### Conclusion:
- Global Maximum: \( 5 \) at \( x = 1 \)
- Global Minimum: \( 1 \) at \( x = 3 \)

Would you like to explore the behavior of such functions further or have more details on this topic?

Here are five related questions:
1. How do you determine global extrema for a function over a closed interval?
2. Can a local maximum also be a global maximum in all cases?
3. What role do endpoints play in finding global extrema?
4. How do second derivatives help identify local extrema?
5. How does the behavior of a function at infinity affect its global extrema?

**Tip**: Always check the function’s behavior at endpoints and infinity to ensure accurate determination of global extrema.

Learn how to determine the global extrema of a function given local extrema. This solution explains how local maximum at (1, 5) and local minimum at (3, 1) can be used to find the global maximum and minimum points using calculus concepts. Suitable for high school students in Grades 10-12.

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