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Student[Calculus1]

 CriticalPoints
 find the critical points of an expression

 Calling Sequence CriticalPoints(f(x), x, opts) CriticalPoints(f(x), x = a..b, opts) CriticalPoints(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 critical points opts - equation(s) of the form numeric=true or false; specify computation options

Description

 • The CriticalPoints(f(x), x) command returns all critical points of f(x) as a list of values.
 • The CriticalPoints(f(x), x = a..b) command returns all critical 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.
 • A critical point is defined as any point at which the derivative is either zero or does not exist.
 • If the expression has an infinite number of critical points, a warning message and sample critical 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 critical 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

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Calculus1}]\right):$
 > $\mathrm{CriticalPoints}\left(3{x}^{2}-x\right)$
 $\left[\frac{{1}}{{6}}\right]$ (1)
 > $\mathrm{CriticalPoints}\left(3{x}^{5}-5{x}^{3}+3,x\right)$
 $\left[{-}{1}{,}{0}{,}{1}\right]$ (2)
 > $\mathrm{CriticalPoints}\left(2{x}^{3}+5{x}^{2}-4x\right)$
 $\left[{-}{2}{,}\frac{{1}}{{3}}\right]$ (3)
 > $\mathrm{CriticalPoints}\left(2{x}^{3}+5{x}^{2}-4x,x=0..1\right)$
 $\left[\frac{{1}}{{3}}\right]$ (4)
 > $\mathrm{CriticalPoints}\left(\frac{{x}^{2}-3x+1}{x},x\right)$
 $\left[{-}{1}{,}{1}\right]$ (5)
 > $\mathrm{CriticalPoints}\left(\frac{{x}^{2}-3x+1}{x},x,\mathrm{numeric}\right)$
 $\left[{-}{1.000000000}{,}{1.000000000}\right]$ (6)