What is a Local Minimum?
A local minimum is a point on a graph where a function reaches a low point relative to the values immediately around it. At this point, the function value is lower than the values of nearby points, even if it’s not the function’s lowest value overall. Local minima often appear in polynomial functions, especially those of degree three or higher, where the curve has enough flexibility to dip and rise multiple times.
Local minima are useful for identifying low points in economic data, physical systems, or optimization problems where you’re looking for the least value within a limited range.
The term local minima is simply the plural form of local minimum. Analysts use the plural form when a function has more than one dip. For example, a graph may have two or more local minima if the curve decreases and increases at several points.
In calculus and algebra, local minima help identify turning points—places where a function changes direction from decreasing to increasing. They are important in curve analysis, optimization strategies, and modeling processes that cycle through ups and downs.
Key Characteristics of Local Minima
- A local minimum occurs at x = a if f(a) is less than the values of f(x) for all x near a.
- It is not necessarily the lowest point on the graph—that would be a global minimum.
- On a smooth curve, a local minimum typically occurs where the slope (derivative) is zero.
- A function can have more than one local minimum or none at all.
Graph Example
In the function f(x) = x³ − 3x² + 2, the graph dips and then rises. This dip occurs at x = 2, where f(x) reaches a local minimum. Even though the function decreases at lower x-values, the value at x = 2 is considered a local minimum because it is lower than all nearby values.
Although the curve continues downward to the left of this point, it does not form another low point near x = 2. Instead, that earlier downward trend is part of the broader shape of the function, and the minimum at x = 2 stands out as a distinct local valley.
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