Change of sign

This article covers sign changes in mathematical functions; To change the accidental in musical notation, see Accidental (music) .

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

Change of sign for continuous functions

Pole with change of sign

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: ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle x_{0}\in (a,b)}$ ${\displaystyle f}$

• Sign changes from plus to minus: there exists a so for all and for all true ${\displaystyle \varepsilon >0}$ ${\displaystyle f(x)>0}$ ${\displaystyle x\in (x_{0}-\varepsilon ,x_{0})}$ ${\displaystyle f(x)<0}$ ${\displaystyle x\in (x_{0},x_{0}+\varepsilon )}$
• Sign change from minus to plus: there exists a so for all and for all true ${\displaystyle \varepsilon >0}$ ${\displaystyle f(x)<0}$ ${\displaystyle x\in (x_{0}-\varepsilon ,x_{0})}$ ${\displaystyle f(x)>0}$ ${\displaystyle x\in (x_{0},x_{0}+\varepsilon )}$

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. ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

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 ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle x_{0}\in (a,b)}$ ${\displaystyle f'(x_{0})=0}$ ${\displaystyle f'}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

• a local maximum when the sign changes from plus to minus ${\displaystyle f'}$
• a local minimum when the sign changes from minus to plus ${\displaystyle f'}$

In the first case the function is for strictly monotonically increasing and for strictly monotonically decreasing, in the second case it is the opposite. ${\displaystyle f}$ ${\displaystyle x<x_{0}}$ ${\displaystyle x>x_{0}}$

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 ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle x_{0}\in (a,b)}$ ${\displaystyle f''(x_{0})=0}$ ${\displaystyle f''}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

• from convex to concave when the sign changes from plus to minus ${\displaystyle f''}$
• from concave to convex when the sign changes from minus to plus ${\displaystyle f''}$

In the first case the derivative is for strictly monotonically increasing and for strictly monotonically decreasing, in the second case it is the opposite. ${\displaystyle f'}$ ${\displaystyle x<x_{0}}$ ${\displaystyle x>x_{0}}$

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 ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle [a,b]}$ ${\displaystyle \alpha ,\beta \in [a,b]}$

${\displaystyle f(\alpha )\cdot f(\beta )\leq 0}$

applies. Even applies

${\displaystyle f(\alpha )\cdot f(\beta )<0}$,

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). ${\displaystyle f}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$

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 ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle x_{0}}$

${\displaystyle f(x)=0}$.

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 ${\displaystyle \alpha \neq \beta }$ ${\displaystyle f(\alpha )\cdot f(\beta )\leq 0}$ ${\displaystyle ([\alpha _{n},\beta _{n}])_{n}}$ ${\displaystyle \alpha _{0}=\alpha }$ ${\displaystyle \beta _{0}=\beta }$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle f(\alpha _{n})\cdot f(\beta _{n})\leq 0}$

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 ${\displaystyle [\alpha _{0},\beta _{0}]}$

${\displaystyle x_{0}=\lim _{n\to \infty }\alpha _{n}=\lim _{n\to \infty }\beta _{n}}$.

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 ${\displaystyle (a_{n})}$ ${\displaystyle (i,i+1)}$

${\displaystyle a_{i}\cdot a_{i+1}<0}$

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

${\displaystyle 2,0,1,0,0,-2,2,1,0,-1}$

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 . ${\displaystyle P}$ ${\displaystyle \sigma (a)}$ ${\displaystyle P}$ ${\displaystyle a}$ ${\displaystyle P}$ ${\displaystyle (a,b]}$ ${\displaystyle \sigma (a)-\sigma (b)}$

See also

• Sign table
• Sign function
• Zero crossing

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

1. ↑ ^{a b} Wolfgang Luh, Karin Stadtmüller: Mathematics for economists . Oldenbourg, 2004, p. 144 .
2. ↑ Hannes Stoppel: Mathematics vividly: bridging course with Maple . Oldenbourg, 2002, p. 26 .
3. ^ Wolfgang Luh, Karin Stadtmüller: Mathematics for economists . Oldenbourg, 2004, p. 150 .
4. ^ ^{A b} Rolf Walter: Introduction to Analysis, Part 1 . de Gruyter, 2007, p. 138 .
5. ^ Günter Scheja, Uwe Storch: Textbook of Algebra, Part 2 . Springer, 1988, p. 112 .