Absolute extremum

Definition

An absolute extremum is the highest or lowest value that a function attains on a given interval. It includes both absolute maximum and absolute minimum values.

5 Must Know Facts For Your Next Test

1. Absolute extrema can occur at critical points or endpoints of a closed interval.
2. To find absolute extrema, evaluate the function at its critical points and endpoints.
3. A function may have more than one absolute extremum on different intervals.
4. The Extreme Value Theorem guarantees that a continuous function on a closed interval has both an absolute maximum and minimum.
5. Absolute extrema are often used in optimization problems to find the best possible outcome under given constraints.

Review Questions

• What is the difference between an absolute extremum and a local extremum?
• How do you determine whether a point is an absolute extremum?
• Why does the Extreme Value Theorem guarantee the existence of an absolute extremum?

Related terms

Critical Point: A point where the derivative of a function is zero or undefined.

Extreme Value Theorem: \text{If a function is continuous on a closed interval } [a, b], \text{ then it has both an absolute maximum and minimum on that interval.}

Local Extremum: \text{A point where the function attains either a local maximum or minimum within some neighborhood around that point.}