The SecondDerivativeTest command returns the classification of the desired point(s) using the second derivative test. Point(s) can either be classified as minima (min), maxima (max), or saddle points (saddle). Alternatively, the Hessian matrix used by the second derivative test can be returned by using the optional argument.

If the optional parameter is given as output = hessian, this command returns the Hessian matrix, if it exists, for the point(s) specified. Alternatively, this parameter can be given in the form output =list where the list contains one or more of the keywords min, max, or saddle.  The default value for this parameter is output = [min, max, saddle].

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\right)\:$

$\mathrm{SecondDerivativeTest}\left(x^2+y^2\,\left[x\,y\right]=\left[0\,0\right]\,\mathrm{output}='\mathrm{hessian}'\right)$

$\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$

$\mathrm{SecondDerivativeTest}\left(x^2+y^2\,\left[x\,y\right]=\left[0\,0\right]\right)$

$\mathrm{LocalMin}=\left[\left[0\,0\right]\right],\mathrm{LocalMax}=\left[\right],\mathrm{Saddle}=\left[\right]$

$\mathrm{SecondDerivativeTest}\left(x^2-2x+8+y^2-4y+z^2+2z\,\left[x\,y\,z\right]=\left[1\,2\,-1\right]\,\mathrm{output}='\mathrm{hessian}'\right)$

$\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$

$\mathrm{SecondDerivativeTest}\left(x^2-2x+8+y^2-4y+z^2+2z\,\left[x\,y\,z\right]=\left[1\,2\,-1\right]\,\mathrm{output}=\left['\min '\,'\mathrm{saddle}'\right]\right)$

$\mathrm{LocalMin}=\left[\left[1\,2\,\mathrm{-1}\right]\right],\mathrm{Saddle}=\left[\right]$