Absolute minimum

Definition

The absolute minimum of a function is the lowest point over its entire domain. It represents the smallest value that the function attains.

5 Must Know Facts For Your Next Test

1. The absolute minimum can occur at a critical point or an endpoint of the function's domain.
2. To find the absolute minimum, evaluate the function at all critical points and endpoints, then compare these values.
3. A continuous function on a closed interval always has an absolute minimum.
4. The first derivative test can help identify critical points where the function changes from decreasing to increasing.
5. Not all functions have an absolute minimum if their domain is not closed or they are not bounded below.

Review Questions

• How do you determine if a point is a potential candidate for an absolute minimum?
• What role do endpoints play in finding the absolute minimum of a function on a closed interval?
• Can a function have more than one absolute minimum? Why or why not?

Related terms

Critical Point: A point on the graph where the derivative is zero or undefined, which may indicate a local extremum.

First Derivative Test: A method used to determine whether a critical point is a local maximum, local minimum, or neither.

Endpoint: A point at either end of the interval for which the function is defined; essential when evaluating for absolute extrema in closed intervals.