FunctionConvexity

1 if f is convex over its whole domain, -1 if concave, 0 if affine, Indeterminate otherwise.

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

FunctionConvexity(f, x)the convexity of f in the real variable x over the whole real line, from the sign of its second derivative.

Typed signatures

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

Domain: Elementary functions

Details

• Same domain-first convention as FunctionMonotonicity -- a domain that isn't all of $\mathbb R$ answers Indeterminate (confirmed against wolframscript), it isn't a decline.
• For a polynomial, reads the sign of the second derivative in closed form; Exp of an affine argument is always convex ($a^2 e^{ax+b}>0$).
• Declines only where the classifier itself declines, or where a polynomial's second derivative is a genuine multi-term polynomial of degree > 2.

Examples

Basic #

A parabola is convex everywhere

live ↗

In$\mathrm{FunctionConvexity}(x^2, x)$Out$1$

Known value · Wolfram 15.0.0, hand cross-checked

A cubic's second derivative changes sign at the inflection point

live ↗

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

Known value · Wolfram 15.0.0, hand cross-checked

Exp is always convex

live ↗

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

Known value · Wolfram 15.0.0, hand cross-checked

Scope #

live ↗

In$\mathrm{Map}(\operatorname{\_1}↦\mathrm{FunctionConvexity}(\operatorname{\_1}, \bigl\lbrack x, y\bigr\rbrack), \bigl\lbrack x^2, x^2+y^2\bigr\rbrack)$Out$\bigl\lbrack\mathrm{FunctionConvexity}(x^2, \bigl\lbrack x, y\bigr\rbrack), \mathrm{FunctionConvexity}(x^2+y^2, \bigl\lbrack x, y\bigr\rbrack)\bigr\rbrack$

Applications #

The parabola opens upward, so it is convex (1)

live ↗

In$\mathrm{FunctionConvexity}(x^2-1, x)$Out$1$

Properties #

1/x's domain isn't all of R, so we decline rather than guess about its (genuinely mixed) shape

live ↗

In$\mathrm{FunctionConvexity}(\frac{1}{x}, x)$Out$\mathrm{FunctionConvexity}(\frac{1}{x}, x)$

About

Examples	6
Checked against	Wolfram Language 3 agree3 differ
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