Question 1

Examine the function   for relative extrema.

Accepted Answer

A)  relative minimum:   
B)  relative maximum:   
C)  relative minimum:   
D)  relative minimum:   
E)  relative maximum:   
A)  relative minimum:   
B)  relative maximum:   
C)  relative minimum:   
D)  relative minimum:   
E)  relative maximum:   C

Question 2

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   .

Accepted Answer

A)  relative maximum 
B)  relative minimum 
C)  saddle point 
D)  insufficient information 
A)  relative maximum 
B)  relative minimum 
C)  saddle point 
D)  insufficient information C

Question 3

Examine the function   for relative extrema and saddle points.

Accepted Answer

A)  saddle point:   ; relative minimum:   
B)  saddle point:   ; relative minimum:   
C)  relative minimum:   ; relative maximum:   
D)  relative minimum:   ; relative maximum:   
E)  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:   ,   A

Question 4

Examine the function   for relative extrema and saddle points.

Accepted Answer

A)  saddle point:   
B)  relative minimum:   
C)  relative maximum:   
D)  relative minimum:   
E)  no critical points 
A)  saddle point:   
B)  relative minimum:   
C)  relative maximum:   
D)  relative minimum:   
E)  no critical points

Question 5

Examine the function   for relative extrema.

Accepted Answer

A)  relative minimum:   
B)  relative minimum:   
C)  relative maximum:   
D)  relative maximum:   
E)  no relative extrema 
A)  relative minimum:   
B)  relative minimum:   
C)  relative maximum:   
D)  relative maximum:   
E)  no relative extrema

Question 6

Find the absolute extrema of   on the region   .

Accepted Answer

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:   
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:

Question 7

Find the absolute extrema of   on the region bounded by the square with vertices

Accepted Answer

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:   
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:

Question 8

Examine the function   for relative extrema and saddle points.

Accepted Answer

A)  saddle point:   
B)  relative minimum:   
C)  relative minimum:   
D)  saddle point:   
E)  saddle point:   
A)  saddle point:   
B)  relative minimum:   
C)  relative minimum:   
D)  saddle point:   
E)  saddle point:

Question 9

List the critical points of the function   for which the Second Partial Test fails.

Accepted Answer

A)  Test fails at   and   . 
B)  Test fails only at   . 
C)  Test fails at   . 
D)  Test fails at   . 
E)  Test fails only at   . 
A)  Test fails at   and   . 
B)  Test fails only at   . 
C)  Test fails at   . 
D)  Test fails at   . 
E)  Test fails only at   .

Question 10

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.

Accepted Answer

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 
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

Question 11

Examine the function   for relative extrema.

Accepted Answer

A)  relative maximum:   
B)  relative minimum:   
C)  relative maximum:   
D)  relative minimum:   
E)  no relative extrema 
A)  relative maximum:   
B)  relative minimum:   
C)  relative maximum:   
D)  relative minimum:   
E)  no relative extrema

Question 12

Examine the function   for relative extrema.

Accepted Answer

A)  relative minimum:   
B)  relative maximum:   
C)  relative maximum:   
D)  relative maximum:   
E)  relative minimum:   
A)  relative minimum:   
B)  relative maximum:   
C)  relative maximum:   
D)  relative maximum:   
E)  relative minimum:

Question 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.

Accepted Answer

A)  relative minimum at   
B)  relative maximum at   
C)  relative maximum at   
D)  relative minimum at   
E)  no relative extrema 
A)  relative minimum at   
B)  relative maximum at   
C)  relative maximum at   
D)  relative minimum at   
E)  no relative extrema

Question 14

Identify any extrema of the function   by recognizing its given form or its form after completing the square.

Accepted Answer

A)  relative maximum:   
B)  relative minimum:   
C)  relative minimum:   
D)  relative maximum:   
E)  relative maximum:   
A)  relative maximum:   
B)  relative minimum:   
C)  relative minimum:   
D)  relative maximum:   
E)  relative maximum:

Question 15

Examine the function   for relative extrema and saddle points.

Accepted Answer

A)  relative minimum:   
B)  relative minimum:   
C)  relative maximum:   
D)  saddle point:   
E)  saddle point:   
A)  relative minimum:   
B)  relative minimum:   
C)  relative maximum:   
D)  saddle point:   
E)  saddle point:

Question 16

Find the absolute extrema of the function   over the triangular region in the xy-plane with vertices   .

Accepted Answer

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   
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

Question 17

Identify any extrema of the function   by recognizing its given form or its form after completing the square.

Accepted Answer

A)  relative maximum:   
B)  relative maximum:   
C)  relative minimum:   
D)  relative minimum:   
E)  relative maximum:   
A)  relative maximum:   
B)  relative maximum:   
C)  relative minimum:   
D)  relative minimum:   
E)  relative maximum:

Exam 13: Section 9: Functions of Several Variables

Examine the function for relative extrema.

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 .

Examine the function for relative extrema and saddle points.

Examine the function for relative extrema and saddle points.

Examine the function for relative extrema.

Find the absolute extrema of on the region .

Find the absolute extrema of on the region bounded by the square with vertices

Examine the function for relative extrema and saddle points.

List the critical points of the function for which the Second Partial Test fails.

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.

Examine the function for relative extrema.

Examine the function for relative extrema.

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.

Identify any extrema of the function by recognizing its given form or its form after completing the square.

Examine the function for relative extrema and saddle points.

Find the absolute extrema of the function over the triangular region in the xy-plane with vertices .

Identify any extrema of the function by recognizing its given form or its form after completing the square.

Section 1: Preparation for Calculus

Section 2: Preparation for Calculus

Section 3: Preparation for Calculus

Section 4: Preparation for Calculus

Section 5: Preparation for Calculus

Section 6: Preparation for Calculus

Section 1: Limits and Their Properties

Section 2: Limits and Their Properties

Section 3: Limits and Their Properties

Section 4: Limits and Their Properties

Section 5 : Limits and Their Properties

Section 1 : Differentiation

Section 2: Differentiation

Section 3: Differentiation

Section 4 : Differentiation

Section 5: Differentiation

Section 6: Differentiation

Section 7: Differentiation

Section 8: Differentiation

Section 1 : Applications of Differentiation

Section 2: Applications of Differentiation

Section 3: Applications of Differentiation

Section 4: Applications of Differentiation

Section 5: Applications of Differentiation

Section 6: Applications of Differentiation

Section 7: Applications of Differentiation

Section 8: Applications of Differentiation

Section 1: Integration

Section 2: Integration

Section 3: Integration

Section 4: Integration

Section 5: Integration

Section 6: Integration

Section 7: Integration

Section 8: Integration

Section 9: Integration

Section 1: Differential Equations

Section 2: Differential Equations

Section 3: Differential Equations

Section 4: Differential Equations

Section 5: Differential Equations

Section 1: Applications of Integration

Section 2: Applications of Integration

Section 3: Applications of Integration

Section 4: Applications of Integration

Section 5: Applications of Integration

Section 6: Applications of Integration

Section 7: Applications of Integration

Section 1: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 2: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 3: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 4: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 5: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 6: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 7: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 8: Integration Techniques, Lhôpitals Rule, and Improper Integrals

Section 1: Infinite Series

Section 2: Infinite Series

Section 3: Infinite Series

Section 4: Infinite Series

Section 5: Infinite Series

Section 6: Infinite Series