$T\left(\mathrm{P}\right)\=\genfrac{}{}{0ex}{}{\left(f_{\mathrm{xx}}{f}_{\mathrm{yy}}-f_{\mathrm{xy}}^2\right)}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{\mathrm{P}}$ 

$T\left(\mathrm{P}\right)\&gt;0$ and $\&lcub;\begin{array}{cc}{f}_{\mathrm{xx}}\left(\mathrm{P}\right)\&gt;0& \Rightarrow f\left(\mathrm{P}\right) local \mathrm{minimum}\\ f_{\mathrm{xx}}\left(\mathrm{P}\right)<0& \Rightarrow f\left(\mathrm{P}\right) local \mathrm{maximum}\end{array}$

$T\left(\mathrm{P}\right)<0\Rightarrow$P is a saddle point

$T\left(\mathrm{P}\right)\&equals;0\Rightarrow$test fails and no conclusion can be drawn

Table 4.8.1   Second-Derivative test for $f\left(x\&comma;y\right)$ 

$H\left(P\right)$ is the Hessian for $f\left(x_1\&comma;\dots \&comma;x_n\right)$ evaluated at P

$S$ is the sequence $1\&comma;Q_1\&comma;\dots \&comma;Q_n$, where $Q_k\ne 0$ is the $k$th principal minor of $H\left(P\right)$

Signs of $S$ strictly alternate ⇒P is a local maximum

Signs of $S$ are all the same  ⇒P is a local minimum

Signs of $S$ neither alternate nor are all the same ⇒P stationary, but not extreme

At least one $Q_k\&equals;0$ ⇒test fails and no conclusion can be drawn

Table 4.8.2   Generalized Second-Derivative test

Table 4.8.3   Remarks on the theoretical framework for the Second-Derivative test

Example 4.8.1

Find and classify the critical (i.e., stationary) points for $f\left(x\&comma;y\right)\&equals;7-3 x^2-5 y^2\&plus;6 x-9 y\&plus;4$.

Example 4.8.2

Find and classify the critical (i.e., stationary) points for $f\left(x\&comma;y\right)\&equals;x y-x^2-y^2-2 x-2 y\&plus;4.$

Example 4.8.3

Find and classify the critical (i.e., stationary) points for $f\left(x\&comma;y\right)\&equals;x y\&plus;2 x-3 y\&plus;1$.

Example 4.8.4

Find and classify the critical (i.e., stationary) points for $f\left(x\&comma;y\right)\&equals;x^3-y^3-3 x y\&plus;4$.

Example 4.8.5

Find and classify the critical (i.e., stationary) points for $f\left(x\&comma;y\right)\&equals;2 x y-3 x^4-5 y^4\&plus;2-y\&plus;7$.

Example 4.8.6

Find and classify the critical (i.e., stationary) points for $x^2-2x\&plus;8\&plus;y^2-4y\&plus;z^2\&plus;2z$.

Example 4.8.7

Line $L_1$ passes through the point $\left(1\&comma;2\&comma;3\right)$ and has direction ${\mathbf{V}}_1\&equals;2 \mathbf{i}-5 \mathbf{j}\&plus;4 \mathbf{k}$. Line $L_2$ passes through the point $\left(3\&comma;-1\&comma;2\right)$ and has direction ${\mathbf{V}}_2\&equals;3 \mathbf{i}\&plus;7 \mathbf{j}-6 \mathbf{k}$. Show that the lines are skew, and find the minimum distance between them. Hint: Parametrize each line with a different parameter and minimize the square of the distance between an arbitrary point on each line.

Example 4.8.8

Find the minimum distance between the point $\left(1\&comma;-2\&comma;3\right)$ and the plane $5 x\&plus;3 y\&plus;2 z\&equals;7$.

Example 4.8.9

If both $\mathbf{U}\&equals;3 \mathbf{i}-5 \mathbf{j}\&plus;7 \mathbf{k}$ and $\mathbf{V}\&equals;6 \mathbf{i}\&plus;\mathbf{j}-13 \mathbf{k}$ are bound to the origin, project U onto V. Hint: Find the minimum distance from the tip of U to the line along V.

Example 4.8.10

Choose $a$ and $b$ so that the line $y\&equals;a x\&plus;b$ minimizes $S\&equals;\sum _{k\&equals;1}^n{\left(a x_k\&plus;b-y_k\right)}^2$, the sum of squares of the deviations from the points $\left(1\&comma;5.3\right)\&comma;\left(3\&comma;8.8\right)\&comma;\left(5\&comma;12.5\right)\&comma;\left(6\&comma;15.4\right)$ to the line. Such a line is called the "least-squares" line.

Example 4.8.11

Obtain formulas for $a$ and $b$ so that the line $y\&equals;a x\&plus;b$ minimizes $S\&equals;\sum _{k\&equals;1}^n{\left(a x_k\&plus;b-y_k\right)}^2$, the sum of squares of the deviations from the points $\left(x_k\&comma;y_k\right)\&comma;k\&equals;1\&comma;\dots \&comma;n$, to the line.

Example 4.8.12

By minimizing the sum of squares of deviations $S\&equals;\sum _{k\&equals;1}^5{\left(f\left(x_k\right)-y_k\right)}^2$ between $f\left(x\right)\&equals;a x^2\&plus;b x\&plus;c$ and the points $\left(1\&comma;3\right)\&comma;\left(2\&comma;8\right)\&comma;\left(3\&comma;15\right)\&comma;\left(4\&comma;30\right)\&comma;\left(5\&comma;38\right)$, obtain the best least-squares quadratic fit to the data.