Maximum and minimum

Local and global maxima and minima for cos(3πx)/x, 0.1≤x≤1.1

In mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighbourhood (local extremum) or on the function domain in its entirety (global extremum).

Definitions

A real-valued function f′ defined on the real line is said to have a local maximum point at the point x^∗, if there exists some ε > 0, such that f(x^∗) ≥ f(x) when |x − x^∗| < ε. The value of the function at this point is called maximum of the function.

On a graph of a function, its local maxima will look like the tops of hills.

Similarly, a function has a local minimum point at x^∗, if f(x^∗) ≤ f(x) when |x − x^∗| < ε. The value of the function at this point is called minimum of the function.

On a graph of a function, its local minima will look like the bottoms of valleys.

A function has a global (or absolute) maximum point at x^∗, if f(x^∗) ≥ f(x) for all x.

Similarly, a function has a global (or absolute) minimum point at x^∗, if f(x^∗) ≤ f(x) for all x.

Any global maximum (minimum) point is also a local maximum (minimum) point; however, a local maximum or minimum point need not also be a global maximum or minimum point.

Terminology: The terms local and global are synonymous with relative and absolute respectively. Also extremum is an inclusive term that includes both maximum and minimum: a local extremum is a local or relative maximum or minimum, and a global extremum is a global or absolute maximum or minimum.

Restricted domains: There may be maxima and minima for a function whose domain does not include all real numbers. A real-valued function, whose domain is any set, can have a global maximum and minimum. There may also be local maxima and local minima points, but only at points of the domain set where the concept of neighborhood is defined. A neighborhood plays the role of the set of x such that |x − x^∗| < ε.

A continuous (real-valued) function on a compact set always takes maximum and minimum values on that set. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above). The neighborhood requirement precludes a local maximum or minimum at an endpoint of an interval. However, an endpoint may still be a global maximum or minimum. Thus it is not always true, for finite domains, that a global maximum (minimum) must also be a local maximum (minimum).

Terminology: The term optimum can replace either one of the terms maximum or minimum, depending on the context. Some optimization problems (see next paragraph) search for a global maximum value while others search for a global minimum value.

Bold text==Finding maxima and minima==

Finding global maxima and minima is the goal of optimization. If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one.

Local extrema can be found by Fermat's theorem, which states that they must occur at critical points. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test.

For any function that is defined piecewise, one finds maxima (or minima) by finding the maximum (or minimum) of each piece separately; and then seeing which one is biggest (or smallest).

Examples

• The function x² has a unique global minimum at x = 0.
• The function x³ has no global or local minima or maxima. Although the first derivative (3x²) is 0 at x = 0, this is an inflection point.
• The function x³/3 − x has first derivative x² − 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 maximum or minimum.
• 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 2 cos(x) − x has infinitely many local maxima and minima, but no global maximum or minimum.
• The function cos(3πx)/x with 0.1 ≤ x ≤ 1.1 has a global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. (See figure at top of page.)
• The function x³ + 3x² − 2x + 1 defined over the closed interval (segment) [−4,2] has two extrema: one local maximum at x = −1−^√15⁄₃, one local minimum at x = −1+^√15⁄₃, a global maximum at x = 2 and a global minimum at x = −4. (See figure at right)

Functions of more variables

For functions of more than one variable, similar conditions apply.

The global maximum is the point at the top.

For example, in the (enlargeable) figure at the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum because of the possibility of a saddle point. For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum.

A counterexample

Counterexample described in the text

However, for identifying global maxima and minima, there are substantial differences between functions of one and several variables. For example, if a differentiable function f defined on the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by reductio ad absurdum). In two and more dimensions, this argument fails, as the function

$f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in \mathbb {R} ,$

shows. Its only critical point is at (0,0), which is a local minimum with f(0,0) = 0. However, it cannot be a global one, because f(4,1) = −11.

See also

• First derivative test
• Second derivative test
• Limit superior and limit inferior
• Mechanical equilibrium
• Extreme value

External links

• Thomas Simpson's work on Maxima and Minima at Convergence