can a hole be a absolute maximum or minimum

2 min read 28-02-2025

can a hole be a absolute maximum or minimum

Can a Hole Be an Absolute Maximum or Minimum? A Mathematical Exploration

The question of whether a hole can represent an absolute maximum or minimum in a function is a fascinating one that delves into the intricacies of calculus and function behavior. The answer, as we'll explore, is nuanced and depends on how we define "hole" and what kind of extrema we're considering.

Understanding Absolute Extrema

Before diving into holes, let's clarify absolute extrema. An absolute maximum is the largest value a function achieves across its entire domain. Similarly, an absolute minimum is the smallest value. These are global extrema – the highest peak and lowest valley, respectively, over the function's entire landscape.

Types of Holes in Functions

Holes in functions, also known as removable discontinuities, arise when a function is undefined at a specific point but could be made continuous by defining its value at that point. This typically occurs due to a common factor in the numerator and denominator of a rational function that cancels out. Consider the function:

f(x) = (x² - 1) / (x - 1)

This function has a hole at x = 1, because the (x-1) term cancels out, leaving f(x) = x + 1 for all x ≠ 1.

Can a Hole Be an Absolute Extremum? The Answer is No (Usually)

A hole, by its nature, represents a point where the function is undefined. Absolute extrema, however, require the function to attain a specific value. Since the function isn't defined at the hole, it cannot have an absolute maximum or minimum at that point.

Example: In our example function, f(x) = (x² - 1) / (x - 1), there's a hole at x = 1. While the limit of f(x) as x approaches 1 is 2, the function doesn't actually reach the value 2 at x = 1 because it's undefined there. Therefore, the hole at x = 1 isn't an absolute extremum.

Exceptions and Considerations

While the typical answer is no, there are nuances:

Domain Restriction: If we explicitly restrict the domain of a function to exclude a neighborhood around the hole, then we might artificially create a situation where a nearby point becomes an absolute extremum. This is more of a manipulation of the domain than a true property of the hole itself.
Piecewise Functions: In a piecewise function where the hole is a boundary between defined pieces, it is possible that the function's value in the adjacent intervals creates an absolute maximum or minimum at one side of the hole.
One-sided Limits: While the function value isn't defined at the hole, we can still consider the one-sided limits. If the limits at the hole show it approaching a value greater than (or less than) all other values of the function, then we can informally say that "around the hole" there's a maximum or minimum. However, it is not formally at the hole itself.

Conclusion: Holes and Extrema

In the standard mathematical context, a hole in a function cannot represent an absolute maximum or minimum. The function must be defined and attain a value at a point for it to be an extremum. While we can discuss the function's behavior near a hole, the hole itself remains a point of discontinuity and undefined value. Therefore, the concept of an extremum at the hole lacks mathematical rigor.