Absolute minimum

Definition

The absolute minimum of a function is the smallest value that the function attains over its entire domain. It is also known as the global minimum.

5 Must Know Facts For Your Next Test

1. The absolute minimum can occur at critical points or endpoints of the 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 $[a, b]$ will always have an absolute minimum.
4. The first derivative test can help identify critical points, where potential minima might occur.
5. Absolute minima are important in optimization problems where you need to find the least value of a function.

Review Questions

• How do you determine if a point is an absolute minimum?
• Why is it necessary to check both critical points and endpoints when finding an absolute minimum?
• What role does the Extreme Value Theorem play in finding absolute minima?

Related terms

Critical Point: A point on a graph where the derivative is zero or undefined; potential location for local or absolute extrema.

First Derivative Test: A method used to determine whether a critical point is a local maximum, local minimum, or neither by analyzing changes in sign of the derivative.

Extreme Value Theorem:

$\text{If } f \text{ is continuous on } [a, b], \text{ then } f \text{ must attain both an absolute maximum and an absolute minimum on } [a, b].$