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$f\=x^2-2x\+8\+y^2-4y\+z^2\+2z$$\stackrel{\text{assign}}{\to }$$$

Find critical points via first principles

$\frac{\partial }{\partial x} f\=0\,\frac{\partial }{\partial y} f\=0\,\frac{\partial }{\partial z} f\=0$

$2x-2\=0\,2y-4\=0\,2z\+2\=0$

$\stackrel{\text{solve}}{\to }$

$\left\{x\=1\,y\=2\,z\=-1\right\}$

Alternate calculation of the critical points

$f$

$x^2\+y^2\+z^2-2x-4y\+2z\+8$

$\stackrel{\text{gradient}}{\to }$

$\stackrel{\text{to list}}{\to }$

$\left[2x-2\,2y-4\,2z\+2\right]$

$\stackrel{\text{equate to 0}}{\to }$

$\left[2x-2\=0\,2y-4\=0\,2z\+2\=0\right]$

$\stackrel{\text{solve}}{\to }$

$\left\{x\=1\,y\=2\,z\=-1\right\}$

Obtain $f\left(1\,2\,-1\right)$ 

$\genfrac{}{}{0ex}{}{f}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=1\,y\=2\,z\=-1}$ = $2$

Obtain the Hessian and determine that it is positive definite

$H≔\mathrm{VectorCalculus}:-\mathrm{Hessian}\left(f\,\left[x\,y\,z\right]\right)$ = $\stackrel{\text{is positive definite?}}{\to }$$\mathrm{true}$$$

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

$f≔\left(x\,y\,z\right)\to x^2-2x\+8\+y^2-4y\+z^2\+2z\:$

Obtain critical points

$\mathrm{solve}\left(\mathrm{Equate}\left(\mathrm{Gradient}\left(f\left(x\,y\,z\right)\,\left[x\,y\,z\right]\right)\,⟨0\,0\,0⟩\right)\right)$

$\left\{x\=1\,y\=2\,z\=-1\right\}$

Apply Sylvester's criterion

$H≔\mathrm{Matrix}\left(3\,3\,\left(i\,j\right)\to {\mathrm{D}}_{i\,j}\left(f\right)\left(1\,2\,-1\right)\right)$

Generate the sequence $1\,Q_1\,\dots \,Q_n$, where the $Q_k$ are the principal minors.

$1\, \mathrm{seq}\left(\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left(H\left(1..k\,1..k\right)\right)\,k\=1..3\right)$

$1\,2\,4\,8$