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The second derivative test

At a critical point where $\nabla f=\vec 0$ , the second derivative test decides whether it is a local minimum, maximum or a saddle point, using a discriminant $D$ built from the second partial derivatives.

D = f_{xx}f_{yy} - f_{xy}^2

the test value

D>0,\ f_{xx}>0:\ \text{minimum}

local minimum

D<0:\ \text{saddle point}

saddle point

Symbols

$D$	discriminant for the second derivative test
$f_{xx},\,f_{yy},\,f_{xy}$	second partial derivatives

Example

$f=3x^2+2y^2$ at the origin: $f_{xx}=6,\,f_{yy}=4,\,f_{xy}=0$ , $D=24>0$ and $f_{xx}>0$ - local minimum.

D<0

always gives a saddle point, whatever the sign of

f_{xx}

Practise partial derivatives for free →

← Partial derivative and gradient · Lagrange multipliers →

Part of Multivariable Calculus: Partial derivatives.