Relative Extremum vs Absolute Extremum: Clarifying the Differences

Understanding the distinctions between relative and absolute extrema is crucial for anyone studying calculus or mathematical functions. These concepts refer to the maximum and minimum values of a function, but within different contexts and scales. Here, we'll delve into the differences and explore multiple examples to provide a clear understanding.

Defining Relative Extrema

A relative extremum, also known as a local extremum, is the maximum or minimum value of a function within a local region or neighborhood. In other words, a function has a relative extremum at a point if the value of the function at that point is larger or smaller than the values of the function in a small open interval around that point. Unlike absolute extrema, relative extrema do not necessarily denote the highest or lowest value over the entire domain of the function.

Defining Absolute Extrema

In contrast, an absolute extremum is the largest or smallest value that a function takes over its entire domain. An absolute maximum is the highest value, while an absolute minimum is the lowest value. These values occur at the boundaries of the function's domain or within the domain itself, depending on the function's behavior.

Examples of Relative and Absolute Extrema

Example 1

Consider the function fx x^2. On the interval [-1, 3], there is a relative minimum at (0, 0), which is also the absolute minimum on the interval. This function has no relative maximum but reaches an absolute maximum of 9 at (3, 9).

Example 2

Consider the function gx x^3 - 3x over the real line -∞ to ∞. This function has a relative minimum at (1, -2) and a relative maximum at (-1, 2). Importantly, this function has no absolute maximum or minimum on this entire domain.

Example 3

Consider the function hx 2x 1 on the interval [-1, 2]. Within this closed interval, there are no relative extrema. However, since the function is continuous over a closed interval, it must have an absolute maximum and an absolute minimum by the Extreme Value Theorem. The absolute minimum occurs at x -1 with a value of -1, and the absolute maximum occurs at x 2 with a value of 5.

Example 4

Consider the function gx x^3 - 3x but restricted to the interval [-3, 2]. Here, the function gx still has two relative extrema. However, neither of these relative extrema is an absolute extremum. The absolute maximum of 12 occurs at x 2, and the absolute minimum of -10 occurs at x -3.

Additional Considerations

It is worth noting that a function can have a relative extremum without having an absolute extremum. This can occur when the function is defined on an open domain or if the function's limits at infinity do not exist.

For instance, the function fx x^2 on the interval (-1, 2) has a local and absolute minimum at x 0 but no absolute maximum. The function fx x^3 - 2x 1 has a local maximum at x -1 and a local minimum at x 1, but it does not have an absolute maximum over the entire domain.

Key Terminology

Relative extrema can be singular or plural. When singular, it is simply called a relative extremum. When plural, we use the term extrema. This is because extremum is the singular form from Latin, similar to how the singular form of datum is data or the singular form of curriculum is curriculum.

Understanding these subtle distinctions is essential for advanced mathematical analysis and problem-solving. By grasping the concepts of relative and absolute extrema, one can accurately model and analyze functions in various fields, including physics, engineering, and economics.