Extreme Value Theorem
Definition
The Extreme Value Theorem states that if a function is continuous on a closed interval $[a, b]$, then it must attain both a maximum and minimum value on that interval. These extreme values can occur at endpoints or critical points within the interval.
5 Must Know Facts For Your Next Test
- If a function is not continuous on $[a, b]$, the Extreme Value Theorem does not apply.
- The theorem guarantees the existence of extrema but does not tell where they occur.
- Endpoints of the interval $[a, b]$ must be considered when finding absolute extrema.
- The theorem applies to both smooth and non-differentiable functions as long as continuity is maintained on $[a, b]$.
- A critical point where the derivative is zero or undefined within $(a, b)$ may be an extreme value.
Review Questions
- What are the conditions required for the Extreme Value Theorem to hold?
- How do you determine where a function's extreme values might occur on a closed interval?
- Why must endpoints be checked when applying the Extreme Value Theorem?
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Related terms
Continuity: A function is continuous if there are no breaks, jumps, or holes in its graph within an interval.
Critical Point: A point in the domain of a function where its derivative is zero or undefined.
Closed Interval: An interval that includes its endpoints, denoted as $[a, b]$.
