Change of sign
In mathematics, a sign change is a change in the sign of the function values of a real function at one point or within an interval . If a continuous real function has a sign change in an interval, it has at least one zero there after the zero set . A differentiable real function has an extremum at one point if its derivative is zero there and its sign changes. Correspondingly, a twice differentiable real function has a point of inflection at a point if its curvature is zero there and its sign changes. Sign changes in real number sequences play an important role in the analysis of the zeros of polynomials .
Sign change at one point
definition
A real function has a sign change at this point if the function values change their sign from there. A distinction is made between the following two cases:
- Sign changes from plus to minus: there exists a so for all and for all true
- Sign change from minus to plus: there exists a so for all and for all true
If the function is continuous, then penetrates function graph of at the location of the x-axis . There is no change in sign if the graph of the function only touches the x-axis at the point . If the function has a vertical asymptote at this point , it is called a pole with a change in sign.
Determination of extremes
In the curve discussion , the so-called sign change criterion provides a sufficient condition for the presence of an extremum at a point. A differentiable real function has an extremum at the point if is and the sign changes at that point . The function then has
- a local maximum when the sign changes from plus to minus
- a local minimum when the sign changes from minus to plus
In the first case the function is for strictly monotonically increasing and for strictly monotonically decreasing, in the second case it is the opposite.
Determination of turning points
Similarly, the sign change criterion can also be used to determine turning points . A twice differentiable real function has a turning point at the point if is and the sign changes at that point . The curvature behavior of the function then changes
- from convex to concave when the sign changes from plus to minus
- from concave to convex when the sign changes from minus to plus
In the first case the derivative is for strictly monotonically increasing and for strictly monotonically decreasing, in the second case it is the opposite.
Change of sign in an interval
definition
A real function has a sign change in the interval if there are two different places for which
applies. Even applies
- ,
so one speaks of a real change of sign. The inequality condition states that the function in the two places and has a different sign (or is equal to zero).
Zero set
If a continuous real function has a sign change in the interval , this function has at least one zero in this interval , that is, a solution of the equation
- .
According to the definition of a sign change, there are digits with in the interval . Now an interval nesting can be constructed with and so that for everyone
applies. To this end, the interval is successively halved and that particular sub-interval is selected for which the inequality condition is retained. The searched zero then results as
- .
A generalization of this as Nullstellensatz or Nullstellensatz of Bolzano (to Bernard Bolzano known) statement is the intermediate value theorem .
use
In numerical mathematics , finite interval nestings are used for the numerical approximation of zeros. In the bisection method and in the regular falsi method , variants of such interval nesting are used in order to approximately determine a zero of a given continuous function in which two positions with different signs are known. In optimization , such interval nesting methods are used to determine the minima or maxima of a given continuously differentiable function, in that the zeros of the first derivative of the function are determined approximately.
Change of sign in a sequence
definition
If a sequence of real numbers, which are all non-zero, then a sign change of this sequence is an index pair for which
applies. The sign changes of any sequence of real numbers are then defined as the sign changes of the partial sequence of the non-zero elements of this sequence. For example, the sequence has
exactly three sign changes.
use
The sign changes in the coefficient sequence of a real polynomial provide information on the number and distribution of the zeros of the associated polynomial function . According to Descartes' sign rule , the number of positive zeros in a real polynomial is equal to or an even natural number smaller than the number of sign changes in its coefficient sequence . Each zero is counted according to its multiplicity.
Sturm chains are another tool for analyzing the zeros of real polynomials . If a polynomial is without multiple zeros and the number of sign changes of the (finite) sequence of function values of the Sturm chain of is at the point , then according to Sturm's rule the number of zeros of in the half-open interval is just the same .
See also
literature
- Wolfgang Luh, Karin Stadtmüller: Mathematics for economists . Oldenbourg, 2004, ISBN 978-3-486-27569-8 .
- Günter Scheja, Uwe Storch: Textbook of Algebra, Part 2 . Springer, 1988, ISBN 3-519-02212-5 .
- Rolf Walter: Introduction to Analysis, Part 1 . de Gruyter, 2007, ISBN 978-3-11-019539-2 .
Individual evidence
- ↑ ^{a } ^{b} Wolfgang Luh, Karin Stadtmüller: Mathematics for economists . Oldenbourg, 2004, p. 144 .
- ↑ Hannes Stoppel: Mathematics vividly: bridging course with Maple . Oldenbourg, 2002, p. 26 .
- ^ Wolfgang Luh, Karin Stadtmüller: Mathematics for economists . Oldenbourg, 2004, p. 150 .
- ^ ^{A } ^{b} Rolf Walter: Introduction to Analysis, Part 1 . de Gruyter, 2007, p. 138 .
- ^ Günter Scheja, Uwe Storch: Textbook of Algebra, Part 2 . Springer, 1988, p. 112 .