Maxima and minima
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In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the function takes the largest (maximum), or smallest (minimum) value either within a given neighbourhood (local extrema), or on the function domain in its entirety (global extrema).
Definitions
A point x* is a local (or relative) maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. On a graph of a function, its local maxima will look like the tops of hills.
A local minimum is a point x* for which f(x*) ≤ f(x) for all x with |x-x*| < ε. On a graph of a function, its local minima will look like the bottoms of valleys.
A global (or absolute) maximum is a point x* for which f(x*) ≥ f(x) for all x. Similarly, a global minimum is a point x* for which f(x*) ≤ f(x) for all x. Any global maximum (minimum) is also a local maximum (minimum); however, a local maximum or minimum need not also be a global maximum or minimum.
The concepts of maxima and minima are not restricted to functions whose domain is the real numbers. One can talk about global maxima and global minima for real-valued functions whose domain is any set. In order to be able to define local maxima and local minima, the function needs to take real values, and the concept of neighborhood must be defined on the domain of the function. A neighborhood then plays the role of the set of x such that |x - x*| < ε.
One refers to a local maximum/minimum as to a local extremum (or local optimum), and to a global maximum/minimum as to a global extremum (or global optimum).
Finding maxima and minima
Finding global maxima and minima is the goal of optimization. For twice-differentiable functions in one variable, a simple technique for finding local maxima and minima is to look for stationary points, which are points where the first derivative is zero. If the second derivative at a stationary point is positive, the point is a local minimum; if it is negative, the point is a local maximum; if it is zero, further investigation is required.
If the function is defined over a bounded segment, one also need to check the end points of the segment.
Examples
- The function [x^2] has a unique global minimum at x = 0.
- The function [x^3] has no global or local minima or maxima. Although the first derivative (3[x^2]) is 0 at x = 0, the second derivative (6x) is also 0.
- The function [x^3]/ 3 - x has first derivative [x^2 - 1] and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at -1 and +1. From the sign of the second derivative we can see that -1 is a local maximum and +1 is a local minimum. Note that this function has no global maxima or minima.
- The function |x| has a global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0.
- The function cos(x) has infinitely many global maxima at 0, ±2π, ±4π, ..., and infinitely many global minima at ±π, ±3π, ... .
- The function 2cos(x) - x has infinitely many local maxima and minima, but no global maxima or minima.
- The function [ x^3 + 3 x^2 - 2 x + 1 ] defined over the closed interval (segment) [-4,2] (see graph) has two extrema: one local maximum in [ x = (-1-\sqrt)/3 ], one local minimum in [ x = (-1+\sqrt)/3 ], a global maximum on x=2 and a global minimum on x=-4.
Functions of more variables
For functions of more variables similar concepts apply, but there is also the saddle point.See also
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