finds the monotonicity of the function f with the variable x over the reals.
finds the monotonicity of f when x is restricted to the domain dom.
gives the monotonicity of f when x is restricted by the constraints cons.
Details and Options
- Monotonicity is also known as increasing, decreasing, non-decreasing, non-increasing, strictly increasing and strictly decreasing.
- By default, the following definitions are used:
+1 non-decreasing, i.e. for all 0 constant, i.e. for all -1 non-increasing, i.e. for all Indeterminate neither non-decreasing nor non-increasing
- The constant function is both non-decreasing and non-increasing.
- With the setting StrictInequalitiesTrue, the following definitions are used:
+1 increasing, i.e. for all -1 decreasing, i.e. for all Indeterminate neither increasing nor decreasing
- 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 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 True whether to require strict monotonicity
- 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)
Monotonicity over unrestricted reals:
A function that is not real valued has Indeterminate monotonicity:
The function is real valued and increasing for positive :
Monotonicity with constraints on the variable:
Strict monotonicity of a function:
is non-decreasing, but not strictly increasing. is strictly increasing:
FunctionMonotonicity gives a conditional answer here:
By default, FunctionMonotonicity may generate conditions on symbolic parameters:
With GenerateConditionsNone, FunctionMonotonicity fails instead of giving a conditional result:
This returns a conditionally valid result without stating the condition:
By default, all conditions are reported:
With GenerateConditionsAutomatic, 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, FunctionMonotonicity computes the non-strict monotonicity:
With StrictInequalitiesTrue, FunctionMonotonicity computes the strict monotonicity:
Ramp[x]+1 is non-decreasing, but is not strictly increasing. Ramp[x]+x is strictly increasing:
Basic Cases (5)
Positive powers are all non-decreasing for the positive reals :
This shows that the whole family is non-decreasing:
In fact, they are all increasing:
Negative powers are non-increasing for the positive reals :
This shows that the whole family is decreasing:
Exponential functions are increasing for and decreasing for :
Trigonometric functions are non-monotonic over the reals:
But over smaller ranges they are monotone:
Combination Cases (5)
The sum of functions with monotonicity has monotonicity :
The sum has the same monotonicity:
The product of non-negative non-decreasing functions is non-decreasing:
Their product is also non-decreasing:
The composition of non-decreasing functions is non-decreasing:
Their compositions are also non-decreasing:
The inverse of an increasing function is increasing:
The inverse is also increasing:
The range of a non-decreasing function on an interval is :
For comparison, compute the range using FunctionRange:
is non-decreasing and bounded from above for :
The limit of at equals the supremum of :
Terms of the series are non-negative, hence the partial sums are increasing:
The partial sums are bounded from above, hence the series converges:
Use Sum to compute the sum of the series:
If is non-negative, then is a non-decreasing function of :
Write a differentiable function as a sum of an increasing function and a decreasing function:
Check whether the functions need to be adjusted by a constant:
CDF is always non-decreasing:
SurvivalFunction is always non-increasing:
Quantile is always non-decreasing in :
Equation Solving and Optimization (2)
If is increasing and continuous in and , then has exactly one root in :
Use Solve to find the root:
Compute the maximum of when is a non-decreasing function:
The maximum value of is and is attained at :
Properties & Relations (2)
The sum and composition of non-decreasing functions are non-decreasing:
The derivative of a non-decreasing function is non-negative:
Use D to compute the derivative:
Use FunctionSign to verify that the derivative is non-negative:
Wolfram Research (2020), FunctionMonotonicity, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionMonotonicity.html.
Wolfram Language. 2020. "FunctionMonotonicity." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionMonotonicity.html.
Wolfram Language. (2020). FunctionMonotonicity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionMonotonicity.html