(a) Sketch the graph of a function on [-1,2] that has an absolute maximum but no local maximum.

step 1 So we want to sketch a graph from negative 1 and 2 that will say have an absolute max but no loc

(a) Sketch the graph of a function on [-1,2] that has an...

Key Concepts

Endpoint Behavior

Endpoint behavior is an important consideration when analyzing functions defined on closed intervals. Endpoints can serve as candidates for absolute extrema even though, by some definitions, they might not qualify as local extrema due to the absence of an open interval around them. Understanding the role of endpoints is essential when constructing examples or counterexamples in optimization problems and when applying the Extreme Value Theorem on closed intervals.

Local Maximum

A local maximum is a point in the domain of a function where the function value is higher than all nearby values, typically within a small open interval around that point. Unlike the absolute maximum, a local maximum does not have to be the highest overall value of the function but is the highest in its immediate neighborhood. This distinction is crucial when analyzing functions, especially when their behavior changes within different regions of the domain.

Absolute Maximum

An absolute maximum, also known as a global maximum, is the highest function value attained over the entire domain under consideration. In the context of a closed interval, the absolute maximum is guaranteed to exist if the function is continuous, based on the Extreme Value Theorem. This concept is central in optimization and analysis because it identifies the best, or optimal, outcome over all possible inputs in the domain.

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