can a hole be a absolute maximum or minimum

2 min read 05-02-2025

can a hole be a absolute maximum or minimum

The question of whether a hole can represent an absolute maximum or minimum in a function is a nuanced one, touching upon the core definitions of these concepts in calculus. The short answer is: no, a hole (removable discontinuity) cannot be an absolute maximum or minimum. However, the explanation requires a closer look at the definitions and behavior of functions.

Understanding Absolute Extrema

Before diving into holes, let's clarify absolute extrema. An absolute maximum is the largest value a function achieves within its domain, while an absolute minimum is the smallest. Crucially, these values must be attained by the function at some point within its domain.

The Role of the Domain

The domain of a function is the set of all possible input values (x-values). A function can only have an absolute maximum or minimum at a point within its domain. A hole, by definition, represents a point where the function is undefined. The function doesn't "exist" at that point; it's missing a value.

Holes and Removable Discontinuities

A hole, also known as a removable discontinuity, occurs when there's a single point missing from the graph of a function. This is often due to a factor that cancels out in the numerator and denominator of a rational function. For instance, consider the function:

f(x) = (x² - 1) / (x - 1)

This function has a hole at x = 1 because (x - 1) is a factor in both the numerator and the denominator. While the function approaches 2 as x approaches 1, it's explicitly undefined at x = 1.

Why Holes Can't Be Extrema

Because a function is undefined at a hole, it cannot achieve an absolute maximum or minimum there. The function simply doesn't have a value at that point to compare to other values in the domain. It might approach a certain value, as in the example above, but it never actually reaches it at the hole itself.

Visualizing the Concept

Imagine the graph of the function with a hole. The graph might approach a particular y-value near the hole, suggesting a potential maximum or minimum. However, there's a gap; the point is not part of the graph. Therefore, another point within the domain might have a slightly higher or lower y-value, making the apparent maximum or minimum at the hole invalid.

What About the Limit?

While the limit of the function as x approaches the hole might indicate a potential maximum or minimum, the limit itself is not the function's value at that point. The limit describes the behavior of the function near the hole, not at the hole itself. The function's value at the hole is undefined.

Conclusion

In summary, a hole in a function, being a point of discontinuity where the function is undefined, cannot represent an absolute maximum or minimum. Although the function's behavior near the hole might suggest a potential extremum, the function does not attain a value at that point, disqualifying it from being a true absolute maximum or minimum. The concepts of limits and absolute extrema must be carefully distinguished.