Extremum

Author

Eric W. Weisstein
January 4, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/Extremum.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Extremum.html.

©2007 Wolfram Research, Inc. except for portions noted otherwise

Infinitely Many Extrema

<<Utilities`Typesetting`

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},Plot[#,{x,-1,1},PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,#},PlotPoints->100]&/@{Cos[1/x],Sin[1/x]}
]],GraphicsSpacing->-.1]

Lots of Extrema

Plot[Sin[Exp[2x+9]],{x,0,1},PlotStyle->Red,PlotPoints->200,AxesLabel->TraditionalForm/@{x,Sin[Exp[2x+9]]}]

v5

pts=Reduce[D[Sin[Exp[2x+9]],x]==0&&0<x<1]

Developer`SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->20000}];

Length[Reduce[pts][[1,1]]]

Floor[((E^11)/π)-(1/2)]-Ceiling[((E^9)/π)-(1/2)]+1

(r=x/.{ToRules[Reduce[D[Sin[Exp[2x+9]],x]==0&&0<x<1/100,x,Reals]]})//Length//Timing

Plot[Sin[Exp[2x+9]],{x,0,.01},
Epilog->{Red,PointSize[.02],{Point/@({#,Sin[Exp[2#+9]]}&/@r)}}]

v6

(r=x/.{ToRules[Reduce[D[Sin[Exp[2x+9]],x]==0&&0<x<1/100,x,Reals]]})//Length//Timing

(r=x/.{ToRules[Reduce[D[Sin[Exp[2x+9]],x]==0&&0<x<1,x,Reals]]})//Length//Timing

v7

(r=x/.{ToRules[Reduce[D[Sin[Exp[2x+9]],x]==0&&0<x<1/100,x,Reals]]})//Length//Timing

(r=x/.{ToRules[Reduce[D[Sin[Exp[2x+9]],x]==0&&0<x<1,x,Reals]]})//Length//Timing