More Than Two Variables

Finding

Find the eigenvalues of

$$\left(\begin{array}{cccc}{f}_{xx}^{\prime \prime }(P_0)& f_{xy}^{\prime \prime }(P_0)& f_{xz}^{\prime \prime }(P_0)& \dots \\ f_{yx}^{\prime \prime }(P_0)& f_{yy}^{\prime \prime }(P_0)& f_{yz}^{\prime \prime }(P_0)& \dots \\ f_{zx}^{\prime \prime }(P_0)& f_{zy}^{\prime \prime }(P_0)& f_{zz}^{\prime \prime }(P_0)& \dots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)$$

Interpreting

• all $\lambda$’s > 0 → min
• all $\lambda$’s < 0 → max
• all $\lambda$’s $\ne 0$ but mixed sign → saddle
• any $\lambda =0$ → no info