13.8 Extrema of Functions of Two Variables 797
13.8 Extrema of Functions of Two Variables
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Identify any extrema of the function by recognizing its
given form or its form after completing the square.
a. relative maximum:
b. relative maximum:
c. relative minimum:
d. relative minimum:
e. relative maximum:
____ 2. Identify any extrema of the function by recognizing its
given form or its form after completing the square.
a. relative maximum:
b. relative minimum:
c. relative minimum:
d. relative maximum:
e. relative maximum:
____ 3. Examine the function for relative extrema.
a. relative minimum:
b. relative maximum:
798 Chapter 13: Functions of Several Variables
c. relative maximum:
d. relative maximum:
e. relative minimum:
____ 4. Examine the function for relative extrema.
a. relative minimum:
b. relative maximum:
c. relative minimum:
d. relative minimum:
e. relative maximum:
____ 5. Examine the function for relative extrema.
a. relative maximum:
b. relative minimum:
c. relative maximum:
d. relative minimum:
e. no relative extrema
____ 6. Examine the function for relative extrema and
saddle points.
a. saddle point:
b. relative minimum:
13.8 Extrema of Functions of Two Variables 799
c. relative minimum:
d. saddle point:
e. saddle point:
____ 7. Examine the function for relative extrema and saddle
points.
a. relative minimum:
b. relative minimum:
c. relative maximum:
d. saddle point:
e. saddle point:
____ 8. Examine the function for relative extrema and saddle
points.
a. saddle point: ; relative minimum:
b. saddle point: ; relative minimum:
c. relative minimum: ; relative maximum:
d. relative minimum: ; relative maximum:
e. saddle points: ,
800 Chapter 13: Functions of Several Variables
____ 9. Examine the function for relative extrema and saddle points.
a. saddle point:
b. relative minimum:
c. relative maximum:
d. relative minimum:
e. no critical points
____ 10. Determine whether there is a relative maximum, a relative minimum, a saddle point,
or insufficient information to determine the nature of the function at the critical point
, if .
a. relative maximum
b. relative minimum
c. saddle point
d. insufficient information
____ 11. List the critical points of the function for which the
Second Partial Test fails.
a. Test fails at and .
b. Test fails only at .
c. Test fails at .
d. Test fails at .
e. Test fails only at .
13.8 Extrema of Functions of Two Variables 801
____ 12. Find the critical points of the function , and
from the form of the function, determine whether a relative maximum or a relative minimum occurs
at each point.
a. relative minimum at
b. relative maximum at
c. relative maximum at
d. relative minimum at
e. no relative extrema
____ 13. Find the critical points of the function , and
from the form of the function, determine whether a relative maximum or a relative minimum occurs
at each point.
a. relative maxima at , , , where are arbitrary
real numbers
b. relative minima at , , , where are arbitrary
real numbers
c. relative minimum at
d. relative maximum at
e. no relative extrema
802 Chapter 13: Functions of Several Variables
____ 14. Find the absolute extrema of on the region bounded
by the square with vertices
a. absolute minimum:
absolute maximum:
b. absolute minimum:
absolute maximum:
c. absolute minimum:
absolute maximum:
d. absolute minimum:
absolute maximum:
e. absolute minimum:
absolute maximum:
____ 15. Find the absolute extrema of the function over the triangular
region in the xy-plane with vertices .
a. absolute maximum: at
absolute minimum: at and along the line
b. absolute maximum: at
absolute minimum: at and along the line
c. absolute maximum: at
absolute minimum: at and along the line
d. absolute maximum: at
absolute minimum: at
e. absolute maximum: at
absolute minimum: at
13.8 Extrema of Functions of Two Variables 803
____ 16. Find the absolute extrema of on the region
.
a. absolute minimum:
absolute maximum:
b. absolute minimum:
absolute maximum:
c. absolute minimum:
absolute maximum:
d. absolute minimum:
absolute maximum:
e. absolute minimum:
absolute maximum:
804 Chapter 13: Functions of Several Variables
13.8 Extrema of Functions of Two Variables
Answer Section
MULTIPLE CHOICE