finds the real sign of the function f with variables x1,x2,… over the reals.
finds the real sign with variables restricted to the domain 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 StrictInequalitiesTrue, 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 xi 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".
Examplesopen allclose all
Basic Examples (3)
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 with constraints on variables:
FunctionSign gives a conditional answer here:
By default, FunctionSign may generate conditions on symbolic parameters:
With GenerateConditionsNone, 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:
Use PerformanceGoal to avoid potentially expensive computations:
The default setting uses all available techniques to try to produce a result:
By default, FunctionSign computes the non-strict sign:
With StrictInequalitiesTrue, FunctionSign computes the strict sign:
is non-negative, but not strictly positive. is strictly positive:
Basic Applications (3)
The graph of lies in the upper half-plane:
The graph of lies in the lower half-plane:
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:
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:
Test whether the limit of is less than :
Prove that the integral is divergent:
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:
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:
Wolfram Research (2020), FunctionSign, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionSign.html.
Wolfram Language. 2020. "FunctionSign." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionSign.html.
Wolfram Language. (2020). FunctionSign. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionSign.html