Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how

FunctionSign

FunctionSign[f,{x₁,x₂,…}]

finds the real sign of the function f with variables x₁,x₂,… over the reals.

FunctionSign[f,{x₁,x₂,…},dom]

finds the real sign with variables restricted to the domain dom.

FunctionSign[{f,cons},{x₁,x₂,…},dom]

gives the sign when variables are restricted by the constraints cons.

Details and Options

Function sign is also known as positive, non-negative, negative, non-positive, strictly positive and strictly negative.
By default, the following definitions are used:

		+1	non-negative, i.e. for all
		0	identically zero, i.e. for all
		-1	non-positive, i.e. for all
		Indeterminate	neither non-negative nor non-positive

The zero function is both non-negative and non-positive.
With the setting StrictInequalitiesTrue, the following definitions are used:
+1 positive, i.e. for all

-1 negative, i.e. for all

Indeterminate neither positive nor negative
Possible values for dom include: Reals, Integers, PositiveReals, PositiveIntegers, etc. The default is Reals.
The function f should be a real-valued function for all x_i in the domain dom that satisfy the constraints cons.
cons can contain equations, inequalities or logical combinations of these.
The following options can be given:

	Assumptions	$Assumptions	assumptions on parameters
	GenerateConditions	True	whether to generate conditions on parameters
	PerformanceGoal	$PerformanceGoal	whether to prioritize speed or quality
	StrictInequalities	False	whether to require a strict sign

Possible settings for GenerateConditions include:
Automatic nongeneric conditions only

True all conditions

False no conditions

None return unevaluated if conditions are needed
Possible settings for PerformanceGoal are "Speed" and "Quality".

Examples

open allclose all

Basic Examples (3)

Find the sign of a function:

Find the sign of a function with variables restricted by constraints:

Find the sign of a function over the integers:

Scope (7)

Univariate functions:

A function that is not real valued has an Indeterminate sign:

The function is real valued and non-negative for positive :

Univariate functions with constraints on the variable:

The strict sign of a function:

is non-negative, but not strictly positive:

Multivariate functions:

Multivariate functions with constraints on variables:

Functions with symbolic parameters:

Options (5)

Assumptions (1)

FunctionSign gives a conditional answer here:

With these assumptions, the function has the opposite sign:

GenerateConditions (2)

By default, FunctionSign may generate conditions on symbolic parameters:

With GenerateConditionsNone, FunctionSign fails instead of giving a conditional result:

This returns a conditionally valid result without stating the condition:

By default, all conditions are reported:

With GenerateConditions->Automatic, conditions that are generically true are not reported:

PerformanceGoal (1)

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

StrictInequalities (1)

By default, FunctionSign computes the non-strict sign:

With StrictInequalitiesTrue, FunctionSign computes the strict sign:

is non-negative, but not strictly positive. is strictly positive:

Applications (14)

Basic Applications (3)

Check the sign of :

The graph of lies in the upper half-plane:

Check the sign of :

The graph of lies in the lower half-plane:

Check the sign of :

The graph of is not contained in either the upper or the lower half-plane:

Show that restricted to is non-negative:

The sum of functions with sign has sign :

The sign of the product of functions is the product of signs:

Calculus (6)

The derivative of a non-decreasing function is non-negative:

If is non-negative, then , for , is non-negative:

A sequence is non-decreasing iff its differences are non-negative:

Sums of non-negative sequences are non-decreasing:

Check the convergence of a non-negative series using d'Alembert's criterion:

Check non-negativity of :

Test whether the limit of is less than :

Prove that the integral is divergent:

Show that :

Show that is non-negative:

Show that the integral of is divergent:

Probability & Statistics (3)

PDF is always non-negative:

CDF is always non-negative:

SurvivalFunction is always non-negative:

Geometry (2)

RegionDistance is always non-negative:

Integral of a non-negative function over a region is non-negative:

Properties & Relations (2)

The sum and product of non-negative functions are non-negative:

A continuous anti-derivative of a non-negative function is non-decreasing:

Use Integrate to compute an anti-derivative:

Use FunctionContinuous to check that the anti-derivative is continuous:

Use FunctionMonotonicity to verify that the anti-derivative is non-decreasing:

Plot the function and the anti-derivative:

Possible Issues (1)

A function must be defined everywhere to have a fixed sign:

Top