FunctionMonotonicity gives a conditional answer here:

Check monotonicity for other values of :

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:

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 StrictInequalitiesTrue, FunctionMonotonicity computes the strict monotonicity:

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

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:

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:

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 :

CDF is always non-decreasing:

SurvivalFunction is always non-increasing:

Quantile is always non-decreasing in :

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: