# Relative Extremum

## Relative Extremum

a maximum or minimum value of a continuous function. More precisely, a function *f*(*x*) that is continuous at a point *x*_{0} has a relative maximum at *x*_{0} if in the domain of *f*(*x*) there exists a neighborhood (*x*_{0} + δ, *x*_{0} – δ) of *x*_{0} such that the inequality *f*(*x*_{0}) ≥ *f*(*x*) is fulfilled at all points of the neighborhood. Similarly, *f*(*x*) has a relative minimum at *x*_{0} if there is a neighborhood such that the inequality *f*(*x*_{0}) ≤ *f*(*x*) holds for all its points.

A relative maximum at *x*_{0} is said to be proper if there exists a neighborhood in which *f*(*x*_{0}) > *f*(*x*) when *x* ≠ *x*_{0}; the term “proper” is applied to a relative minimum at *x*_{0} if there is a neighborhood in which *f*(*x*_{0}) < *f*(*x*) when *x* ≠ *x*_{0}. 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.

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 *x*_{0}, it must be continuous at *x*_{0}; in addition, the derivative *f*(*x*_{0}) 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 *x*_{0} the derivative is positive to the left of *x*_{0} and negative to the right of *x*_{0}, then *f*(*x*) has a relative maximum at *x*_{0}; (2) if *f*′(*x*) is negative to the left of *x*_{0} and positive to the right, then *f*(*x*_{0}) is a relative minimum. If, however, *f′(x)* does not change sign in the neighborhood of *x*_{0}, then *f*(*x*) does not have an extremum at *x*_{0}; examples of such points are *D, E*, and Fin Figure 1.

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

If *f*(*x*) has a relative maximum at *x*_{0} and if *f*(*x*_{0}) ≥ *f*(*x*) for all *x* ≠ *x*_{0} in the domain of the function, then *f*(*x*_{0}) is called an absolute maximum. If *f*(*x*) has a relative minimum at *x*_{0} and if *f*(*x*_{0}) ≤ *f*(*x*) for all *x* ≠ *x*_{0}, then *f*(*x*_{0}) 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*.

Suppose that in some neighborhood of the point *M*(*x*_{0}, *y*_{0}) 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 also*CONDITIONAL RELATIVE EXTREMUM.)

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

### REFERENCE

Il’in, V. A., and E. G. Pozniak.*Osnovy matematicheskogo analiza*, 3rd ed., part 1. Moscow, 1971.