Multiple Choice A function that is continuous on the...

Multiple Choice A function that is continuous on the interval ________________ is guaranteed to have both an absolute maximum and an absolute minimum.
(a) $(a, b)$
(b) $(a, b]$
(c) $[a, b)$
(d) $[a, b]$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Closed and Bounded Intervals

A closed interval, typically denoted by [a, b], is one that includes its endpoints. The bounded nature means that the interval is finite in extent. This closed and finite structure is essential because it ensures that the function has a compact domain, a necessary condition for many theorems in calculus, including the Extreme Value Theorem.

Absolute Extrema

Absolute maximum and minimum (also known as global extrema) refer to the highest and lowest values, respectively, that a function takes on over a given interval. These values are important in optimization problems and are guaranteed to exist for continuous functions on closed intervals.

Extreme Value Theorem

This theorem states that if a function is continuous over a closed and bounded interval, then it is guaranteed to have both an absolute maximum and an absolute minimum on that interval. It underpins how continuity and domain restrictions work together to ensure that no extreme values are missed.

Continuity on a Closed Interval

Continuity of a function over a closed interval means that the function does not have any breaks, jumps, or asymptotes on that interval. This continuity is crucial because, without any gaps or holes, the function's behavior is predictable enough to allow the application of the Extreme Value Theorem.

Transcript

Please give Ace some feedback