AP Calculus AB and BC
Local vs Absolute Extrema
A local extremum is the largest or smallest value compared only to nearby points. An absolute extremum is the largest or smallest across the entire interval. Local extrema occur only at critical points; absolute extrema occur at a critical point or at an endpoint.
Local extremum
Use when: The question asks where the function turns around, or asks for a relative maximum or minimum.
Absolute extremum
Use when: The question asks for the largest or smallest value on a specified closed interval.
Side by side
| Local | Absolute | |
|---|---|---|
| Compared against | Nearby points only | Every point on the interval |
| Can occur at an endpoint | No | Yes |
| Found using | First or second derivative test | The candidates test |
| How many can exist | Several | One maximum value, one minimum value |
Every absolute extremum on a closed interval is either a local extremum or an endpoint value, which is exactly why the candidates test evaluates the function at every critical point and both endpoints and then compares.
The Extreme Value Theorem is what guarantees absolute extrema exist at all, and it needs continuity on a closed interval. On an open interval a function can approach a value without ever attaining it.
The mistake
Comparing derivative values when hunting an absolute extremum. Critical points are located with , but the comparison that decides the answer uses .
Frequently asked questions
Can an endpoint be a local maximum?
Under the usual convention, no. A local extremum needs points on both sides to compare against, so an endpoint can only hold an absolute extremum.
In the CED: Unit 5: Analytical Applications