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

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