Fermat's Stationary Point Theorem

Author

Eric W. Weisstein
August 14, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Calculus/FermatsStationaryPointTheorem.nb.

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Figure

At a differentiable local minimum or maximum, the tangent is horizontal. The stationary inflection point illustrates why the converse is false.

Module[
{
blue=Blue,
epilog,
neutral=LightDarkSwitched[Black,White],
red=Red
},
epilog={Directive[blue,AbsoluteThickness[2]],Line[{{-0.65,0},{0.65,0}}],Directive[red,PointSize[0.022]],Point[{0,0}]};GraphicsRow[{Plot[(x^2),{x,-1.2,1.2},PlotStyle->Directive[neutral,AbsoluteThickness[2]],AxesStyle->neutral,AxesOrigin->{0,0},PlotRange->{{-1.2,1.2},{-1.8,1.8}},Epilog->epilog,PlotLabel->Style["local minimum",15,neutral,FontFamily->"Times"]],Plot[-(x^2),{x,-1.2,1.2},PlotStyle->Directive[neutral,AbsoluteThickness[2]],AxesStyle->neutral,AxesOrigin->{0,0},PlotRange->{{-1.2,1.2},{-1.8,1.8}},Epilog->epilog,PlotLabel->Style["local maximum",15,neutral,FontFamily->"Times"]],Plot[(x^3),{x,-1.2,1.2},PlotStyle->Directive[neutral,AbsoluteThickness[2]],AxesStyle->neutral,AxesOrigin->{0,0},PlotRange->{{-1.2,1.2},{-1.8,1.8}},Epilog->epilog,PlotLabel->Style["stationary inflection point",15,neutral,FontFamily->"Times"]]}]]