FunctionMonotonicity

FunctionMonotonicity[f,x]

finds the monotonicity of the function f with the variable x over the reals.

FunctionMonotonicity[f,x,dom]

finds the monotonicity of f when x is restricted to the domain dom.

FunctionMonotonicity[{f,cons},x,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 StrictInequalitiesTrue, 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".

Examples

open allclose all

Basic Examples (3)

Find the monotonicity of a function:

Find the monotonicity of a function with the variable restricted by constraints:

Find the monotonicity of a function over the integers:

Scope (5)

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:

Functions with symbolic parameters:

Options (5)

Assumptions (1)

FunctionMonotonicity gives a conditional answer here:

Check monotonicity for other values of :

GenerateConditions (2)

By default, FunctionMonotonicity may generate conditions on symbolic parameters:

With GenerateConditionsNone, 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 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, FunctionMonotonicity computes the non-strict monotonicity:

With StrictInequalitiesTrue, FunctionMonotonicity computes the strict monotonicity:

Ramp[x]+1 is non-decreasing, but is not strictly increasing. Ramp[x]+x is strictly increasing:

Applications (19)

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:

is non-decreasing but not increasing:

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:

Calculus (4)

Prove that has a limit for :

is non-decreasing and bounded from above for :

The limit of at equals the supremum of :

Prove convergence 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:

Test monotonicity of and :

Probability (3)

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:

Compute the maximum of :

The maximum value of is and is attained at :

For comparison, compute the maximum directly:

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:

Plot the function and the derivative:

Possible Issues (1)

A function must be defined everywhere to be monotonic:

Top

FunctionMonotonicity

Details and Options

Examples

Basic Examples (3)

Scope (5)