Definition

1. Noun:
   • The most extreme possible amount or value: The greatest or least possible degree, quantity, or intensity of something.
   • The point located farthest from the middle of something: The endpoint or the point of greatest distance within a set or on a curve.

Usage

• As a noun, "extremum" is a technical term primarily used in mathematics, physics, and formal analysis. It refers to a maximum or minimum value.
• It is often used in its plural form, extrema, to refer to both maximum and minimum points collectively.

Examples

• Noun:
  • In calculus, finding the extremum of a function is a fundamental problem. (Finding the maximum or minimum value.)
  • The data shows an extremum at this point, indicating a peak in the signal. (The most extreme value.)
  • The algorithm is designed to locate all local extrema of the waveform. (All maximum and minimum points.)

Advanced Usage

• Local extremum: A point that is a maximum or minimum value within a specific, limited range, but not necessarily the absolute highest or lowest point overall.
  • The function has a local extremum at x=2.
• Global extremum (or absolute extremum): The absolute highest maximum or lowest minimum value of a function over its entire domain.
  • Identifying the global extremum is crucial for optimization problems.

Variants and Related Words

• Extreme (adj/n): (adj) Reaching a high or the highest degree; very great. (n) Either of two abstract things that are as different from each other as possible.
  • The expedition faced extreme weather conditions.
• Extremity (n): The furthest point or limit of something; an extreme degree.
  • The northern extremity of the island.
• Maximum (n/adj): (n) The greatest or highest amount possible or attained. (adj) As great, high, or intense as possible or permitted.
• Minimum (n/adj): (n) The least or smallest amount or quantity possible, attainable, or required. (adj) Smallest or lowest.

Synonyms

• Peak (n): The point of highest activity, quality, or achievement.
• Trough (n): A low point; the opposite of a peak in this context.
• Vertex (n): In mathematics, the highest point; the top or apex, often synonymous with a maximum point on a parabola.
• Endpoint (n): A point that marks the end of a line segment or interval.

Notes

• "Extremum" is a formal, technical word. In everyday language, words like "peak," "bottom," "maximum," or "minimum" are more common.
• In mathematics, the process of finding extrema is called optimization.

Noun

1. the most extreme possible amount or value
   • voltage peak
2. the point located farthest from the middle of something