Absolute maximum

Definition

The absolute maximum of a function is the highest value that the function attains over its entire domain. It represents the peak point on the graph of the function.

5 Must Know Facts For Your Next Test

1. The absolute maximum can occur at a critical point or at an endpoint of the domain.
2. To find the absolute maximum, evaluate the function at all critical points and endpoints, then compare these values.
3. A function may have more than one local maximum but only one absolute maximum over its domain.
4. If a function is continuous on a closed interval $[a, b]$, it must have both an absolute maximum and minimum according to the Extreme Value Theorem.
5. The first derivative test and second derivative test are helpful tools in identifying whether a critical point is an absolute maximum.

Review Questions

• How do you determine if a critical point is an absolute maximum?
• Explain why a continuous function on a closed interval always has an absolute maximum.
• What methods can be used to find the absolute maximum of a function?

Related terms

Critical Point: A point where the first derivative of a function is zero or undefined, indicating potential maxima, minima, or saddle points.

Local Maximum: A value where the function reaches its highest point within a certain interval around that value but not necessarily over its entire domain.

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