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.
congrats on reading the definition of absolute maximum. now let's actually learn it.
The absolute maximum can occur at a critical point or at an endpoint of the domain.
To find the absolute maximum, evaluate the function at all critical points and endpoints, then compare these values.
A function may have more than one local maximum but only one absolute maximum over its domain.
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.
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.}