Find the extremas of $f(x,y,z)=xyz-x^2-y^2-z^2$

Question

Find the extremas of $f(x,y,z)=xyz-x^2-y^2-z^2$.

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After some calculation, $Df(x,y,z)=0$ for and only for $$ \\p_1=(0,0,0), p_2=(2,2,2),p_3=(-2,-2,2),p_4=(-2,2,-2),p_5=(2,-2,-2)\ \\ H(f)=\begin{vmatrix} -2 &z &y \\ z & -2 &x \\ y & x &-2 \end{vmatrix} \ $$ So this test helps only for the maxima $p_1$. About the other critical points, I've tried to calculate $f(x,y,z)-f(x+\varepsilon ,y,z)$ but both for positive and negative $\varepsilon$ the sign remains the same.

Answer 1

Take the point $p_2$. Then your matrix is$$\begin{bmatrix}-2&2&2\\2&-2&2\\2&2&-2\end{bmatrix},$$whose eigenvalues are $-4$ (twice) and $2$. So, $p_2$ is a saddle point of $f$. Apply the same approach to the other critical points.