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.
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Absolute extrema can occur at critical points or endpoints of a closed interval.
To find absolute extrema, evaluate the function at its critical points and endpoints.
A function may have more than one absolute extremum on different intervals.
The Extreme Value Theorem guarantees that a continuous function on a closed interval has both an absolute maximum and minimum.
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.}