Use the definition of a local maximum to explicitly argue...

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Key Concepts

Local Maximum Definition

A local maximum of a function is a point where the function value is at least as high as the values of the function at all points in a nearby open interval. This definition is central in analysis because it formalizes what it means for a function to achieve a peak in a small region of its domain, rather than globally. When using this definition, one must show that there is some interval around the point in question where every other point's function value does not exceed the value at that point.

Neighborhood (Open Interval)

The concept of a neighborhood, or an open interval around a point, is crucial in the definition of local extrema. It refers to a set of points surrounding a particular point whose values are compared to the point in question. In the context of a local maximum, one examines whether every function value in this vicinity is less than or equal to the value at the given point. If there is no such interval because other points arbitrarily close have higher values, then the point cannot be considered a local maximum.

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