FunctionMonotonicity

1 if f is increasing over its whole domain, -1 if decreasing, 0 if constant, Indeterminate otherwise.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFunctionMonotonicity!↗

FunctionMonotonicity(f, x)whether f is monotonic in the real variable x over the whole real line.

Typed signatures

Type	Package	Applies
(expression, symbol) -> expression	analytic	always

Domain: Elementary functions

Details

• Matches Wolfram's own convention, confirmed against wolframscript, that a proper (non-Reals) domain already answers Indeterminate -- FunctionMonotonicity(Sqrt(x), x) and FunctionMonotonicity(Log(x), x) are Indeterminate, not declined, because their domain isn't all of $\mathbb R$, not because this file can't analyze them.
• For a polynomial with domain $\mathbb R$, the sign of its derivative is read off in closed form -- constant sign (possibly touching zero at one point) gives 1 or -1, a proven sign change gives Indeterminate. Exp of an affine argument is always 1 or -1, by the sign of the argument's slope.
• Declines only when the classifier itself declines (see FunctionDomain) or when a polynomial's derivative is a genuine multi-term polynomial of degree > 2 (its sign isn't determined here).

Examples

Basic #

A cubic is increasing everywhere -- its derivative $3x^2$ never goes negative

live ↗

In$\mathrm{FunctionMonotonicity}(x^3, x)$Out$1$

Known value · Wolfram 15.0.0, hand cross-checked

A parabola turns around at its vertex

live ↗

In$\mathrm{FunctionMonotonicity}(x^2, x)$Out$\mathrm{FunctionMonotonicity}(x^2, x)$

Known value · Wolfram 15.0.0, hand cross-checked

Exp of an affine argument is always monotonic

live ↗

In$\mathrm{FunctionMonotonicity}(e^{x}, x)$Out$1$

Known value · Wolfram 15.0.0, hand cross-checked

Scope #

live ↗

In$\mathrm{Map}(()↦\mathrm{FunctionMonotonicity}(\operatorname{\_1}, x), \bigl\lbrack\mathrm{Ramp}(x)+1, x+\mathrm{Ramp}(x)\bigr\rbrack)$Out$\bigl\lbrack\mathrm{FunctionMonotonicity}(\mathrm{Ramp}(x)+1, x), \mathrm{FunctionMonotonicity}(x+\mathrm{Ramp}(x), x)\bigr\rbrack$

Properties #

Sqrt's domain isn't all of R, so Wolfram's own convention is Indeterminate here -- confirmed against wolframscript

live ↗

In$\mathrm{FunctionMonotonicity}(\sqrt{x}, x)$Out$\mathrm{FunctionMonotonicity}(\sqrt{x}, x)$

Known value · Wolfram 15.0.0, hand cross-checked

About

Examples	5
Checked against	Wolfram Language 2 agree3 differ
Crosswalk	Wolfram Language