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.
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The absolute maximum can be found by evaluating the function at critical points and endpoints of its domain.
A function may have more than one absolute maximum if it attains the same highest value at multiple points.
To determine an absolute maximum, one must first find the derivative to locate critical points where the slope is zero or undefined.
The absolute maximum can occur within an interval or at a boundary point, depending on whether the function is defined at those endpoints.
If a function is continuous on a closed interval, it will always have both an absolute maximum and minimum according to the Extreme Value Theorem.
Review Questions
What steps are necessary to find the absolute maximum of a given function?
Can a function have multiple points that qualify as its absolute maximum? Explain why or why not.
How does the Extreme Value Theorem help in finding an absolute maximum?
Related terms
Critical Point: A point on a graph where the derivative is zero or undefined, which could be indicative of local maxima, minima, or saddle points.
Extreme Value Theorem: A theorem stating that if a function is continuous on a closed interval, then it must attain both an absolute maximum and minimum within that interval.
Local Maximum: The highest value of a function within a specific region around a certain point; not necessarily higher than values outside this region.