Find the point on the plane 3x + 2y + z = 6 that is nearest the origin.

Among all the points on the graph of

$$z = 10 - x ^ { 2 } - y ^ { 2 }$$

that lie above the plane x + 2y + 3z = 0, find the point farthest from the plane.

Find three numbers whose sum is 9 and whose sum of squares is a minimum.

Find the point on the plane z = x + y + 1 closest to the point P = (1, 0, 0). Hint: Minimize the square of the distance.

Find the shortest distance from the point (2, 0, -3) to the plane x + y + z = 1.

Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.

A rectangular box without a top (a topless box) is to be made from 12 $\mathrm { ft } ^ { 2 }$ of cardboard. Find the maximum volume of such a box.

Prove the Cayley-Hamilton Theorem using Jordan form.

Find the points on the surface y^2=9+xz that are closest to the origin.

Find the distance between the planes 5x - 3y + z = 2 and 5x - 3y + z = -3.

Use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these. $f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } + 2 x \right) }$

Find for each geometric series described.

$$S _ { n } = 4118 , a _ { n } = 128 , r = \frac { 2 } { 3 }$$

Find the minimum distance from the point (2, -1, 1) to the plane x + y - z = 2.

Use matrix multiplication to check that the inverse of a general $2 \times 2$ matrix is given by

$$\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}{d} & {-b} \\ {-c} & {a}\end{array}\right]$$

(provided $a d-b c \neq 0$).