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

1. If a function is not continuous on $[a, b]$, the Extreme Value Theorem does not apply.
2. The theorem guarantees the existence of extrema but does not tell where they occur.
3. Endpoints of the interval $[a, b]$ must be considered when finding absolute extrema.
4. The theorem applies to both smooth and non-differentiable functions as long as continuity is maintained on $[a, b]$.
5. 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?

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 $[a, b]$.