There's some debate over this issue. Many definitions indicate that endpoints canNOT be local (relative) extrema. However, there are also many definitions that indicate otherwise, so you should check with your teacher on the specific definition to be used for your class.blue_soda025 said:I'm kind of confused about whether or not an endpoint of an interval could also be a local extreme value. According to my textbook, it's true, but on the review sheets, the answers never included endpoints as local extrema. So can endpoints be local extrema?
A local extrema with endpoint included is a point on a curve or surface that represents either the maximum or minimum value of a function within a specific interval, including the endpoints of the interval.
To find local extrema with endpoint included, you can use the first or second derivative test. The first derivative test involves setting the first derivative of the function equal to zero and solving for the critical points. The second derivative test involves evaluating the second derivative at the critical points to determine the nature of the extrema.
Including endpoints when finding local extrema is important because it allows us to accurately determine the maximum or minimum value of a function within a specific interval. Without including endpoints, we may miss critical points and obtain incorrect results.
Yes, a local extrema with endpoint included can also be a global extrema. A global extrema is the maximum or minimum value of a function on its entire domain, while a local extrema is the maximum or minimum value within a specific interval. If the interval includes the endpoints of the function's domain, then the local extrema with endpoint included is also the global extrema.
Real-life applications of finding local extrema with endpoint included include optimizing production processes in manufacturing, determining the most profitable price point for a product or service, and finding the best route to minimize travel time in transportation. It can also be used in fields such as economics, physics, and engineering for optimization problems.