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Student[Calculus1]

 InflectionPoints
 find the inflection points of an expression

 Calling Sequence InflectionPoints(f(x), x, opts) InflectionPoints(f(x), x = a..b, opts) InflectionPoints(f(x), a..b, opts)

Parameters

 f(x) - algebraic expression in variable 'x' x - name; specify the independent variable a, b - algebraic expressions; specify restricted interval for inflection points opts - equation(s) of the form numeric=true or false; specify computation options

Description

 • 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 $-10$ and $10$ if they are not provided. By default, the value is false.

Examples

 > $\mathrm{with}\left({\mathrm{Student}}_{\mathrm{Calculus1}}\right):$
 > $\mathrm{InflectionPoints}\left(3{x}^{2}-x\right)$
 $\left[\right]$ (1)
 > $\mathrm{InflectionPoints}\left(3{x}^{5}-5{x}^{3}+3,x\right)$
 $\left[{-}\frac{\sqrt{{2}}}{{2}}{,}{0}{,}\frac{\sqrt{{2}}}{{2}}\right]$ (2)
 > $\mathrm{InflectionPoints}\left(3{x}^{5}-5{x}^{3}+3,x=-0.5..0.5\right)$
 $\left[{0}\right]$ (3)
 > $\mathrm{InflectionPoints}\left(2{x}^{3}+5{x}^{2}-4x,x\right)$
 $\left[{-}\frac{{5}}{{6}}\right]$ (4)
 > $\mathrm{InflectionPoints}\left(2{x}^{3}+5{x}^{2}-4x,x,\mathrm{numeric}\right)$
 $\left[{-0.8333333333}\right]$ (5)
 > $\mathrm{InflectionPoints}\left(\frac{{x}^{2}-3x+1}{x}\right)$
 $\left[\right]$ (6)