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 StrictlyMonotonicFunctionQ (1.0.0) current version: 2.0.0 »

Source Notebook

Check if a function is strictly monotonic

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["StrictlyMonotonicFunctionQ"][f,x]

gives True if f is strictly monotonic with respect to x.

ResourceFunction["StrictlyMonotonicFunctionQ"][f,x,property]

restricts the test to the direction dir.

Details and Options

The direction dir can be any of Automatic, "Increasing" or "Decreasing" and defaults to Automatic. When dir is Automatic, StrictlyMonotonicQ returns True if f is strictly increasing or strictly decreasing.
ResourceFunction["StrictlyMonotonicFunctionQ"] expects f to be a univariate expression in terms of x, similar to what might be used in Plot.
ResourceFunction["StrictlyMonotonicFunctionQ"] returns True based on the definition of f being strictly monotonic; that is, f is strictly increasing on an interval A if for all {x,y}∈A such that x<y, it is the case that f[x]<f[y].

Examples

Basic Examples (2) 

Compute whether an expression is strictly monotonic:

In[1]:=
ResourceFunction["StrictlyMonotonicFunctionQ"][x^3, x]
Out[1]=

Determine if an expression is increasing:

In[2]:=
ResourceFunction["StrictlyMonotonicFunctionQ"][x^5, x, "Increasing"]
Out[2]=

Properties and Relations (3) 

Note that a function with regions where it is constant is not strictly increasing:

In[3]:=
f = Piecewise[{{x, x <= 0}, {0, 0 < x < 1}, {x - 1, x > 1}}];
In[4]:=
ResourceFunction["StrictlyMonotonicFunctionQ"][f, x]
Out[4]=

For more information on the monotonicity of the input expression, use the resource function FunctionMonotonicity:

In[5]:=
ResourceFunction["FunctionMonotonicity"][f, x]
Out[5]=

Such a curve is only monotonic, not strictly monotonic:

In[6]:=
ResourceFunction["MonotonicFunctionQ"][f, x]
Out[6]=

Publisher

Wolfram|Alpha Math Team

Version History

  • 2.0.0 – 23 March 2023
  • 1.0.0 – 29 September 2020

Related Resources

Author Notes

To view the full source code for StrictlyMonotonicFunctionQ, run the following code:

In[1]:=
FileNameJoin[ ReplacePart[ FileNameSplit[FindFile["ResourceFunctionHelpers`"]], -1 -> "FunctionMonotonicityConcavity.wl"]] // SystemOpen

License Information