Function Repository Resource:

InflectionPoints

Source Notebook

Find the inflection points of a function of one variable

Contributed by: Wolfram|Alpha Math Team

ResourceFunction["InflectionPoints"][expr, x]

computes the inflection points of the expression expr with respect to variable x.

ResourceFunction["InflectionPoints"][{expr,constraint},x]

computes the inflection points expr, subject to the given condition constraint on x.

ResourceFunction["InflectionPoints"][…, "Properties"]

computes the inflection points of a function, along with function properties at these points.

Details and Options

Inflection points are returned as a list of rules for the independent variable x. When the "Properties" directive is invoked, inflection points are listed along with properties such as "rising", "falling", "stationary" and "non-stationary".

For functions with a repeating pattern of inflection points, ResourceFunction["InflectionPoints"] returns results in terms of one or more undetermined constants,

which can take any integer value.

In ResourceFunction["InflectionPoints"][{expr,constraint},x], the constraint argument should be an inequality involving x.

Examples

Basic Examples (4)

Find the inflection points of a cubic function:

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Plot the function and its inflection points found above:

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Repeat the calculation, classifying inflection points:

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Find and classify the inflection points of a polynomial function:

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Plot the function and its inflection points:

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Find and classify the inflection points of another polynomial function:

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Plot the function and its inflection points:

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Find and classify the inflection points of a trigonometric function:

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Plot the function and a single cycle's worth of its inflection points:

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Options (2)

Find and classify the inflection points of a function, specifying the range of the independent variable to include:

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$func = E^Sin[x^3] - 1; inflPts = ResourceFunction["InflectionPoints"][{func, -2 <= x <= 2}, x, "Properties"]$

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Plot the function and its inflection points:

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Properties and Relations (3)

InflectionPoints will sometimes return results in terms of Root objects:

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Applying N to these outputs converts to an ordinary numeric result:

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Plot the function and its inflection points:

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For functions with a repeating pattern of inflection points, InflectionPoints returns results in terms of one or more undetermined constants, which can take any integer value:

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