The InflectionPoints(f(x), x) command returns all inflection points of f(x) as a list of values.

The InflectionPoints(f(x), x = a..b) command returns all inflection points of f(x) in the interval [a,b] as a list of values.

If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.

An inflection point is defined as any point at which the sign of the second derivative changes.

If the expression has an infinite number of inflection points, a warning message and sample inflection points are returned.

The opts argument can contain the following equation that sets computation options.

numeric = true or false

Whether to use numeric methods (using floating-point computations) to find the inflection points of the expression. If this option is set to true, the points a and b must be finite and are set to $\mathrm{-10}$ and $10$ if they are not provided. By default, the value is false.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right)\:$

$\mathrm{InflectionPoints}\left(3x^2-x\right)$

$\mathrm{InflectionPoints}\left(3x^5-5x^3+3\,x\right)$

$\left[-\frac{\sqrt{2}}{2}\,0\,\frac{\sqrt{2}}{2}\right]$

$\mathrm{InflectionPoints}\left(3x^5-5x^3+3\,x=-0.5..0.5\right)$

$\left[0\right]$

$\mathrm{InflectionPoints}\left(2x^3+5x^2-4x\,x\right)$

$\left[-\frac{5}{6}\right]$

$\mathrm{InflectionPoints}\left(2x^3+5x^2-4x\,x\,\mathrm{numeric}\right)$

$\left[\mathrm{-0.8333333333}\right]$

$\mathrm{InflectionPoints}\left(\frac{x^2-3x+1}{x}\right)$