LOCAL MINIMUM Meaning and
Definition
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A local minimum refers to a point in a mathematical function or a graph at which the value of the function is smaller than at any nearby points, but it may not be the absolute smallest value in the entire function or graph. In other words, when examining a curve or surface, a local minimum represents a low point in the immediate vicinity or in a small interval around that particular point, but it could still exist in the presence of even lower values elsewhere.
Mathematically, we can determine a local minimum by analyzing the derivatives of the function. At a local minimum, the first derivative of the function is zero, and the second derivative is positive. This suggests that the function is decreasing as we move away from the point, confirming its relative minimum nature.
It is important to note that a local minimum does not guarantee the absolute minimum of a function, as the absolute minimum refers to the lowest value across the entire function or graph. Therefore, a function can have multiple local minima, and the absolute minimum may be different from these local minima.
The concept of local minimum finds applications in various fields, including optimization problems, machine learning algorithms, and data analysis. By identifying local minima, researchers can determine and compare different low points within a system to make informed decisions and evaluate performance.
Etymology of LOCAL MINIMUM
The word "local minimum" is derived from the combination of two terms: "local" and "minimum".
The term "local" originates from the Latin word "locus", meaning "place" or "site". It implies a restricted or limited scope, referring to something that occurs or is applicable within a particular area or region.
The word "minimum" comes from the Latin word "minimus", which means "smallest" or "least". In mathematics and optimization theory, a minimum refers to the smallest or lowest possible value that a function can attain within a given range or domain.
Combining these two words, "local minimum" refers to the lowest point or value of a function within a specific region or small area, indicating that it is not necessarily the absolute or global lowest point, but only the lowest within its immediate vicinity.
