Relative Extremum

a maximum or minimum value of a continuous function. More precisely, a function f(x) that is continuous at a point x₀ has a relative maximum at x₀ if in the domain of f(x) there exists a neighborhood (x₀ + δ, x₀ – δ) of x₀ such that the inequality f(x₀) ≥ f(x) is fulfilled at all points of the neighborhood. Similarly, f(x) has a relative minimum at x₀ if there is a neighborhood such that the inequality f(x₀) ≤ f(x) holds for all its points.

A relative maximum at x₀ is said to be proper if there exists a neighborhood in which f(x₀) > f(x) when x ≠ x₀; the term “proper” is applied to a relative minimum at x₀ if there is a neighborhood in which f(x₀) < f(x) when x ≠ x₀. The function shown in Figure 1 has a proper maximum at point A; although the function’s value at point B is a minimum, it is not a proper minimum.

Figure 1

A point at which a function has a relative maximum is called a maximum point, and a point at which it has a relative minimum is called a minimum point. In order for a function f(x) to have a relative extremum at some point x₀, it must be continuous at x₀; in addition, the derivative f(x₀) must either be equal to zero, as at point A in Figure 1, or be nonexistent, as at point C. The following statements present sufficient conditions for relative extrema: (1) if in some neighborhood of x₀ the derivative is positive to the left of x₀ and negative to the right of x₀, then f(x) has a relative maximum at x₀; (2) if f′(x) is negative to the left of x₀ and positive to the right, then f(x₀) is a relative minimum. If, however, f′(x) does not change sign in the neighborhood of x₀, then f(x) does not have an extremum at x₀; examples of such points are D, E, and Fin Figure 1.

Suppose f(x) has n successive derivatives at x₀. If f′(x₀) = f′(x₀) = . . . = f^{(n – 1)}(x₀) = 0 and f⁽ⁿ⁾(x₀) ≠ 0, then f(x) does not have a relative extremum at x₀ if n is odd and does have a relative extremum if n is even. The extremum is a minimum if f⁽ⁿ⁾(x₀) > 0 and a maximum if f′⁽ⁿ⁾(x₀) < 0.

If f(x) has a relative maximum at x₀ and if f(x₀) ≥ f(x) for all x ≠ x₀ in the domain of the function, then f(x₀) is called an absolute maximum. If f(x) has a relative minimum at x₀ and if f(x₀) ≤ f(x) for all x ≠ x₀, then f(x₀) is said to be an absolute minimum.

A relative extremum of a function of several variables is defined in much the same way as a relative extremum of a function of a single variable. A necessary condition for a function of several variables to have a relative extremum is that the first-order partial derivatives be equal to zero or be nonexistent. For example, in Figure 2,a they are equal to zero at point M, and in Figure 2,b they do not exist at M.

Figure 2

Suppose that in some neighborhood of the point M(x₀, y₀) there exist continuous first-and second-order partial derivatives of the function f(x, y). If at M we have and , then f(x, y) has a relative extremum at M. The extremum is a maximum if and a minimum if . A relative extremum does not exist at M if Δ < 0; in this case M is called a saddle point (see Figure 2,c).

The sufficient conditions for a relative extremum of a function of many variables reduce to the quadratic form

being positive or negative definite; here, a_ik is the value of

at the point in question. (See alsoCONDITIONAL RELATIVE EXTREMUM.)

The term “extremum” is also used in the study of maximum and minimum values of functionals in the calculus of variations.