(A) How many local minimum points does f have in ? (The
answer is an integer).

(B) How many local maximum points does f have in ?

(C) How many saddle points does f have in ?

A) 0

B) 0

C) 1

The work

2.

Consider the function f (x, y) = xsin(y).

(A) How many local minima does f have in ?

(B) How many local maxima does f have in ?

(C) How many saddle points does f have in ?

A) 0

B) 0

C) ∞

The work

3.

Consider the function . Find and classify all critical points of the function.

The critical point with the smallest x-coordinate is
(, ) Classification:

Critical Point = (-2, -6)
Classification: saddle point

The work

4.

Consider the function
Find and classify all critical points of the function.

The critical point with the smallest x-coordinate is
(, ) Classification:

Critical point: ( -4, -2)
Classification: Local Maximum

The work

5.

Consider the function
Find and classify all critical points of the function.

The critical point with the smallest x-coordinate is
(, ) Classification:

-2

Critical point: ( 1, 6)
Classification: Local Maximum

The work

6.

Consider the function
Find and classify all critical points of the function.

The critical point with the smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:

The work

7.

Consider the function
Find and classify all critical points of the function.

The critical point with the smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:

Critical point: ( 0, 0)
Classification: Saddle Point

Critical point: ( 0, 8)
Classification: Saddle Point

Critical point: ( 6, 4)
Classification: Local Maximum

Critical point: ( 12, 0)
Classification: Saddle Point

Critical point: ( 12, 8)
Classification: Saddle Point

The work

8.

Find the absolute minimum and absolute maximum of
on the closed triangular region with vertices
(0,0), (10, 0) and (10, 13).

List the maximum/minimum values as well as the point(s) at which they
occur.

Minimum value:
Occurs at (,)

Maximum value:
Occurs at (,)

Minimum value: -56 at ( 10, 0)

Maximum value: 74 at ( 10, 13)

The work

9.

Find the absolute maximum and absolute minimum of the function
on the region

Absolute minimum value:
attained at (, ) .

Absolute maximum value:
attained at (, ), (, ).

Absolute minimum value: -128
attained at ( -4, 0) .

Absolute maximum value: 256
attained at ( 0, -4), ( 0, 4).

The work

10.

Find the absolute maximum and minimum of the function on the domain .

Absolute minimum value:
attained at (, ).

Absolute maximum value:
attained at (, ).

Absolute minimum value: -28
attained at ( 0, 7).

Absolute maximum value: 3
attained at ( 1, 2).

The work

11.

Find the coordinates of the point (x, y, z) on the plane
z = 2 x + 3 y + 1 which is closest to the origin.
x =
y =
z =

x = -0.143
y = -0.2143
z = 0.07143

The work

12.

Find the point(s) on the surface which are closest to
the point (10, 11, 0).

(6,8,-7), (6,8,7)

The work

13.

Find three positive real numbers whose sum is 73 and whose product is
a maximum.
, ,

73/3, 73/3, 73/3

The work

14.

Find the dimensions of the rectangular box having the largest volume and surface area 122 square units.
, ,.

4.509, 4.509, 4.509

The work

15.

Find the volume of the largest rectangular box in the first octant
with three faces in the coordinate planes, and one vertex in the plane
.
Largest volume is