# 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 , 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

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## 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 , 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 , 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 , 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: