Deck 14: Directional Derivatives, Gradients, and Extrema

Question 1

Let $f ( x , y ) = x ^ { 2 } y + y ^ { 2 }$ and $\mathbf { u } = \frac { 1 } { 2 } \mathbf { i } + \frac { \sqrt { 3 } } { 2 } \mathbf { j }$ Then the directional derivative of f at (-1, 2) in the direction of $\mathbf { U }$ is

A) $\frac { - 2 + 5 \sqrt { 3 } } { 2 }$
B) $\frac { 4 - 5 \sqrt { 3 } } { 2 }$
C) $\frac { - 4 + 5 \sqrt { 3 } } { 2 }$
D) $\frac { - 4 - 5 \sqrt { 3 } } { 2 }$
E) $\frac { 4 + 5 \sqrt { 3 } } { 2 }$

Answer 1

$\frac { - 4 + 5 \sqrt { 3 } } { 2 }$

Question 2

Let $f ( x , y ) = x e ^ { y } + y e ^ { x }$ Then the directional derivative of f at (0,0) in the direction of the unit vector $\mathbf { U }$ which makes an angle of $\frac { \pi } { 6 }$ from the positive x-axis is

A) $\frac { \sqrt { 3 } + 1 } { 2 }$
B) $\frac { \sqrt { 3 } - 1 } { 2 }$
C) $- \frac { \sqrt { 3 } - 1 } { 2 }$
D) $- \frac { \sqrt { 3 } - 1 } { 2 }$ .
E) $\frac { \sqrt { 3 } + 3 } { 2 }$

Answer 2

$\frac { \sqrt { 3 } + 1 } { 2 }$

Question 3

Let $f ( x , y ) = e ^ { x y } + 2 x ^ { 2 } y ^ { 2 }$ . Then the directional derivative of f at (0,1) in the direction of the unit vector $\mathbf { U }$ which is parallel to $\mathbf { v } = - 5 \mathbf { i } + 12 \mathbf { j }$ is

A) $\frac { 5 } { 12 }$
B) $- \frac { 5 } { 12 }$
C) $\frac { 5 } { 13 }$
D) $- \frac { 5 } { 13 }$
E) $- \frac { 7 } { 13 }$

Answer 3

$- \frac { 5 } { 13 }$

Question 4

Let $f ( x , y ) = \ln \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Then the directional derivative of f at (3,4) in the direction of the unit vector $\mathbf { U }$ which is parallel to $\mathbf { v } = 5 \mathbf { i } + 12 \mathbf { j }$ is

A) $\frac { \sqrt { 3 } + 1 } { 2 }$
B) $\sqrt { 15 }$
C) $- \frac { 7 } { 13 }$
D) $- \frac { 5 } { 13 }$
E) $\frac { 63 } { 325 }$

Question 5

Let $f ( x , y , z ) = x ^ { 2 } y - x y z ^ { 2 }$ . Then the directional derivative of f at $P = ( 0,1,2 )$ in the direction of the unit vector $\mathbf { U }$ which is parallel to $\overrightarrow { P Q }$ where $Q = ( 1,4,3 )$ is

A) $\frac { \sqrt { 3 } + 1 } { 2 }$
B) $- \frac { 4 } { \sqrt { 26 } }$
C) $\frac { - 4 + 5 \sqrt { 3 } } { 2 }$
D) $- \frac { 5 } { 13 }$
E) $\frac { 78 } { 325 }$

Question 6

Let $f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + z ^ { 2 } x$ Then the directional derivative of f at $P = ( 1,2 , - 1 )$ in the direction of the unit vector $\mathbf { u }$ which is parallel to $\overrightarrow { P Q }$ where $Q = ( 2,0,1 )$ is

A) $\sqrt { 5 }$
B) $\frac { 12 } { \sqrt { 5 } }$
C)5
D) $- \frac { 5 } { 13 }$
E) $\frac { 78 } { 325 }$

Question 7

The gradient of $f ( x , y , z ) = x ^ { 2 } y - x y z ^ { 2 }$ at (0, 1, 2) is

A) $- 4 \mathbf { k }$
B) $5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k }$
C) $- 4 \mathbf { i }$
D) $5 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$
E) $- 4 \mathbf { j }$

Question 8

The gradient of $f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + z ^ { 2 } x$ at (1, 2, -1) is

A) $- 5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$
B) $5 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k }$
C) $5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$
D) $- 5 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$
E) $5 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$

Question 9

The gradient of $f ( x , y , z ) = z \tan ^ { - 1 } \left( \frac { y } { x } \right)$ at (1, 1, 3) is

A) $\frac { 3 } { 2 } \mathbf { i } - \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$
B) $- \frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$
C) $\frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$
D) $- \frac { 3 } { 2 } \mathbf { i } - \frac { 3 } { 2 } \mathbf { j } + \frac { \pi } { 4 } \mathbf { k }$
E) $- \frac { 3 } { 2 } \mathbf { i } + \frac { 3 } { 2 } \mathbf { j } - \frac { \pi } { 4 } \mathbf { k }$

Question 10

The gradient of $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ at (3, 0, -4) is

A) $\frac { 3 } { 5 } \mathbf { i } - \frac { 4 } { 5 } \mathbf { k }$
B) $\frac { 3 } { 5 } \mathbf { i } + \frac { 4 } { 5 } \mathbf { k }$
C) $- \frac { 3 } { 5 } \mathbf { i } - \frac { 4 } { 5 } \mathbf { k }$
D) $- \frac { 3 } { 5 } \mathbf { i } + \frac { 4 } { 5 } \mathbf { k }$
E) $\frac { 3 } { 5 } \mathbf { i } - \frac { 4 } { 5 } \mathbf { j }$

Question 11

The gradient of $f ( x , y , z ) = 2 x ^ { 3 } + x y ^ { 2 } + z ^ { 2 } x$ at (1, 1, 1) is

A) $- 8 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
B) $8 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k }$
C) $8 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }$
D) $8 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }$
E) $8 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$

Question 12

The gradient of $f ( x , y , z ) = y ^ { 2 } + z ^ { 2 } - 4 x z$ at (-2, 1, 3) is

A) $- 12 \mathbf { i } - 2 \mathbf { j } + 14 \mathbf { k }$
B) $- 12 \mathbf { i } + 2 \mathbf { j } + 14 \mathbf { k }$
C) $- 12 \mathbf { i } + 2 \mathbf { j } - 14 \mathbf { k }$
D) $12 \mathbf { i } + 2 \mathbf { j } + 14 \mathbf { k }$
E) $12 \mathbf { i } + 2 \mathbf { j } - 14 \mathbf { k }$

Question 13

Let $f ( x , y ) = x y ^ { 2 } + x ^ { 2 }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( - 1,2 )$ is

A) $\sqrt { 5 }$
B) $2 \sqrt { 3 }$
C) $2 \sqrt { 5 }$
D) $\sqrt { 35 }$
E)5

Question 14

Let $f ( x , y ) = 4 x y + x ^ { 2 } + 2 y ^ { 2 }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 2,1 )$ is

A) $4 \sqrt { 13 }$
B) $3 \sqrt { 13 }$
C) $2 \sqrt { 13 }$
D) $\sqrt { 13 }$
E)13

Question 15

Let $f ( x , y ) = 2 x e ^ { y } + 3 y e ^ { x }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 0,0 )$ is

A) $4 \sqrt { 13 }$
B) $3 \sqrt { 13 }$
C) $2 \sqrt { 13 }$
D) $\sqrt { 13 }$
E)13

Question 16

Let $f ( x , y ) = 2 x \ln y$ Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 4,1 )$ is

A)24
B)4
C)16
D)8
E)2

Question 17

Let $f ( x , y ) = \frac { x y } { x ^ { 2 } + y ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( - 1,2 )$ is

A)3
B) $\frac { 3 \sqrt { 5 } } { 25 }$
C) $\frac { 3 } { 5 }$
D) $\frac { 1 } { 3 }$
E)5

Question 18

Let $f ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 2,4 )$ is

A) $\sqrt { 3 }$
B)2
C)3
D)1
E)5

Question 19

Let $f ( x , y , z ) = z \tan ^ { - 1 } \left( \frac { y } { x } \right)$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 1,1,3 )$ is

A) $\frac { \sqrt { 72 + \pi ^ { 2 } } } { 2 }$
B) $\frac { \sqrt { 72 + \pi ^ { 2 } } } { 4 }$
C) $\frac { \sqrt { 72 - \pi ^ { 2 } } } { 2 }$
D) $\frac { \sqrt { 72 - \pi ^ { 2 } } } { 4 }$
E) $\frac { \sqrt { 72 + \pi ^ { 2 } } } { 3 }$

Question 20

Let $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ . Then the maximum value of the directional derivative $D _ { \mathrm { u } } f ( 3,4,0 )$ is

A)5
B)3
C)2
D)1
E) $\frac { 1 } { 3 }$

Question 21

An equation of the tangent plane to the surface $16 z = 4 x ^ { 2 } + y ^ { 2 }$ at (2, 4, 2) is

A) $- 2 x + y - 2 z = 4$
B) $2 x - y + 2 z = 4$
C) $2 x + y + 2 z = 4$
D) $2 x + y - 2 z = 4$
E) $2 x - y - 2 z = 4$

Question 22

The symmetric equations of the normal line to the surface $16 z = 4 x ^ { 2 } + y ^ { 2 }$ at (2, 4, 2) are

A) $\frac { x - 2 } { 4 } = y - 4 = \frac { z - 2 } { - 2 }$
B) $\frac { x - 2 } { 2 } = y - 4 = \frac { z - 2 } { - 4 }$
C) $\frac { x - 2 } { 2 } = y - 4 = \frac { z - 2 } { - 2 }$
D) $\frac { x - 2 } { 2 } = y - 4 = \frac { z - 2 } { 2 }$
E) $\frac { x - 2 } { - 2 } = y - 4 = \frac { z - 2 } { - 2 }$

Question 23

An equation of the tangent plane to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) is

A) $8 x - 4 y + 12 z = 52$
B) $8 x - 4 y - 12 z = 52$
C) $8 x + 4 y + 12 z = 52$
D) $- 8 x - 4 y + 12 z = 52$
E) $- 8 x + 4 y + 12 z = 52$

Question 24

The symmetric equations of the normal line to the surface $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 26$ at (1, -2, 3) are

A) $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { 3 }$
B) $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { 2 }$
C) $\frac { x - 1 } { - 3 } = y + 2 = \frac { z - 3 } { - 2 }$
D) $\frac { x - 1 } { 2 } = y + 2 = \frac { z - 3 } { - 3 }$
E) $\frac { x - 1 } { - 2 } = y + 2 = \frac { z - 3 } { - 3 }$

Question 25

An equation of the tangent plane to the surface $3 z = x ^ { 2 } + y ^ { 2 } - 2$ at (-2, -4, 6) is

A) $4 x - 8 y + 3 z = - 22$
B) $4 x + 8 y + 3 z = - 22$
C) $4 x + 8 y - 3 z = - 22$
D) $4 x - 8 y - 3 z = - 22$
E) $- 4 x + 8 y + 3 z = - 22$

Question 26

The symmetric equations of the normal line to the surface $3 z = x ^ { 2 } + y ^ { 2 } - 2$ at (-2, -4, 6) are

A) $\frac { x + 2 } { 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { 3 }$
B) $\frac { x + 2 } { - 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { 3 }$
C) $\frac { x + 2 } { 4 } = \frac { y + 4 } { - 8 } = \frac { z - 6 } { 3 }$
D) $\frac { x + 2 } { 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { - 3 }$
E) $\frac { x + 2 } { - 4 } = \frac { y + 4 } { 8 } = \frac { z - 6 } { - 3 }$

Question 27

An equation of the tangent plane to the surface $y = e ^ { x } \cos z$ at (1, e, 0) is

A) $y = - e x + z$
B) $y = e x - z$
C) $y = e x + z$
D) $y = - e x$
E) $y = e x$

Question 28

The symmetric equations of the normal line to the surface $y = e ^ { x } \cos z$ at (1, e, 0) are

A) $\frac { x - 1 } { 2 e } = y - e , z = 0$
B) $\frac { x - 1 } { - 2 e } = y - e , z = 0$
C) $\frac { x - 1 } { 3 e } = y - e , z = 0$
D) $\frac { x - 1 } { e } = y - e , z = 0$
E) $\frac { x - 1 } { - e } = y - e , z = 0$

Question 29

An equation of the tangent plane to the surface $x ^ { 2 } = 12 y$ at (-6, 3, 1) is

A) $x - y = 7$
B) $x + y = 7$
C) $x + y = 9$
D) $x - y = 9$
E) $- x + y = 9$

Question 30

The symmetric equations of the normal line to the surface $x ^ { 2 } = 12 y$ at (-6, 3, 1) are

A) $\frac { x + 6 } { 2 } = y - 3 , z = 1$
B) $x + 6 = y - 3 , z = 1$
C) $x + 6 = \frac { y - 3 } { 2 } , z = 1$
D) $\frac { x + 6 } { 3 } = y - 3 , z = 1$
E) $x + 6 = \frac { y - 3 } { 3 } , z = 1$

Question 31

An equation of the tangent plane to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at (1, 1, 4) is

A) $2 x - 2 y + z = 8$
B) $2 x + 2 y + z = 8$
C) $2 x + 2 y - z = 8$
D) $2 x - 2 y - z = 8$
E) $- 2 x + 2 y + z = 8$

Question 32

The symmetric equations of the normal line to the surface $\sqrt { x } + \sqrt { y } + \sqrt { z } = 4$ at (1, 1, 4) are

A) $\frac { x - 1 } { 2 } = 4 ( y - 1 ) = z - 4$
B) $\frac { x - 1 } { - 4 } = 4 ( y - 1 ) = z - 4 = 4 ( y - 1 ) = z - 4$
C) $\frac { x - 1 } { 4 } = 4 ( y - 1 ) = z - 4$
D) $- 4 ( x - 1 ) = 4 ( y - 1 ) = z - 4$
E) $( x - 1 ) = ( y - 1 ) = 2 ( z - 4 )$

Question 33

An equation of the tangent plane to the surface $z x ^ { 2 } - x y ^ { 2 } - y z ^ { 2 } = 18$ at (0, -2, 3) is

A) $4 x - 9 y + 12 z = 54$
B) $4 x - 9 y - 12 z = 54$
C) $- 4 x + 9 y + 12 z = 54$
D) $- 4 x - 9 y - 12 z = 54$
E) $- 4 x - 9 y + 12 z = 54$

Question 34

The symmetric equations of the normal line to the surface $z x ^ { 2 } - x y ^ { 2 } - y z ^ { 2 } = 18$ at (0, -2, 3) are

A) $\frac { x } { - 4 } = \frac { y + 2 } { - 9 } = \frac { z - 3 } { 12 }$
B) $\frac { x } { - 4 } = \frac { y + 2 } { 9 } = \frac { z - 3 } { 12 }$
C) $\frac { x } { 4 } = \frac { y + 2 } { - 9 } = \frac { z - 3 } { 12 }$
D) $\frac { x } { 4 } = \frac { y + 2 } { 9 } = \frac { z - 3 } { 12 }$
E) $\frac { x } { 9 } = \frac { y + 2 } { - 4 } = \frac { z - 3 } { 12 }$

Question 35

An equation of the tangent plane to the surface $x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } + z ^ { \frac { 2 } { 3 } } = 14$ at (-8, 27, 1) is

A) $- 3 x + 2 y - 6 z = - 84$
B) $3 x + 2 y + 6 z = - 84$
C) $3 x - 2 y - 6 z = - 84$
D) $3 x + 2 y - 6 z = - 84$
E) $3 x - 2 y + 6 z = - 84$

Question 36

The symmetric equations of the normal line to the surface $x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } + z ^ { \frac { 2 } { 3 } } = 14$ at (-8, 27, 1) are

A) $\frac { x + 8 } { 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { - 6 }$
B) $\frac { x + 8 } { - 3 } = \frac { y - 27 } { 2 } = \frac { z - 1 } { - 6 }$
C) $\frac { x + 8 } { 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { 6 }$
D) $\frac { x + 8 } { 3 } = \frac { y - 27 } { 2 } = \frac { z - 1 } { 6 }$
E) $\frac { x + 8 } { - 3 } = \frac { y - 27 } { - 2 } = \frac { z - 1 } { 6 }$

Question 37

An equation of the tangent plane to the surface $x ^ { 2 } - 3 y ^ { 2 } - 4 z ^ { 2 } = 2$ at (3, 1, 1) is

A) $- 3 x - 3 y - 4 z = 2$
B) $3 x + 3 y + 4 z = 2$
C) $3 x - 3 y + 4 z = 2$
D) $3 x - 3 y - 4 z = 2$
E) $3 x + 3 y - 4 z = 2$

Question 38

The symmetric equations of the normal line to the surface $x ^ { 2 } - 3 y ^ { 2 } - 4 z ^ { 2 } = 2$ at (3, 1, 1) are

A) $\frac { x - 3 } { - 3 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { - 4 }$
B) $\frac { x - 3 } { 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { 4 }$
C) $\frac { x - 3 } { 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { - 4 }$
D) $\frac { x - 3 } { - 3 } = \frac { y - 1 } { - 3 } = \frac { z - 1 } { 4 }$
E) $\frac { x - 3 } { 3 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 4 }$

Question 39

The set of points on $4 x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 50$ at which the tangent plane is parallel to the xy-plane is

A) $\{ ( 0,0 , - 5 ) , ( 0,0,5 ) \}$
B) $\{ ( 0,0,5 ) \}$
C) $\{ ( 0,0 , - 5 ) \}$
D) $\{ ( 0,0 , - 3 ) \}$
E) $\{ ( 0,0,3 ) \}$

Question 40

The set of points on $z = x ^ { 2 } - 3 x y + y ^ { 2 }$ at which the tangent plane is parallel to the xy-plane is

A) $\{ ( 0,0,3 ) \}$
B) $\{ ( 0,0,2 ) \}$
C) $\{ ( 0,0,1 ) \}$
D) $\{ ( 0,0,0 ) \}$
E) $\{ ( 0,0,5 ) \}$

Question 41

Let $f ( x , y ) = x ^ { 3 } + y ^ { 2 } - 6 x ^ { 2 } + y - 1$ . Then f has a relative minimum at

A) $( 4 , - 2 )$
B) $\left( - 4 , \frac { 1 } { 2 } \right)$
C) $\left( - 4 , - \frac { 1 } { 2 } \right)$
D) $\left( 4 , \frac { 1 } { 2 } \right)$
E) $\left( 4 , - \frac { 1 } { 2 } \right)$

Question 42

Let $f ( x , y ) = \frac { 1 } { x } - \frac { 64 } { y } + x y$ . Then f has a relative maximum at

A) $\left( - \frac { 1 } { 4 } , 16 \right)$
B) $\left( \frac { 1 } { 4 } , 16 \right)$
C) $\left( \frac { 1 } { 4 } , 32 \right)$
D) $\left( - \frac { 1 } { 4 } , - 16 \right)$
E) $\left( \frac { 1 } { 4 } , - 16 \right)$

Question 43

Let $f ( x , y ) = 4 x y ^ { 2 } - 2 x ^ { 2 } y - x$ . Then the set of saddle point(s) of f is

A) $\{ ( 0 , - 2,0 ) \}$
B) $\{ ( 0 , - 2,0 ) , ( 0,2,0 ) \}$
C) $\left\{ \left( 0 , \frac { 1 } { 2 } , 0 \right) \right\}$
D) $\left\{ \left( 0 , - \frac { 1 } { 2 } , 0 \right) , \left( 0 , \frac { 1 } { 2 } , 0 \right) \right\}$
E) $\left\{ \left( 0 , - \frac { 1 } { 2 } , 0 \right) \right\}$

Question 44

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2$ . Then f has a relative minimum at

A) $( 1,1 )$
B) $( - 1,1 )$
C) $( 1 , - 1 )$
D) $( - 1 , - 1 )$
E) $\left( 1 , \frac { 1 } { 2 } \right)$

Question 45

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2$ . Then f has a relative maximum at

A) $( 1,3 )$
B) $( - 1 , - 3 )$
C) $( 1 , - 3 )$
D) $( - 1,3 )$
E) $\left( - 1 , \frac { 1 } { 3 } \right)$

Question 46

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } + 3 y ^ { 2 } - 3 x - 9 y + 2$ . Then the set of saddle points of f is

A) $\{ ( 1 , - 3,27 ) \}$
B) $\{ ( - 1,1 , - 1 ) \}$
C) $\{ ( 1 , - 3,27 ) , ( - 1,1 , - 1 ) \}$
D) $\{ ( 1 , - 3,27 ) , ( - 1,1,1 ) \}$
E) $\{ ( 1 , - 3,27 ) , ( 1,1 , - 1 ) \}$

Question 47

Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$ . Then f has a relative maximum at

A) $\left( \frac { \pi } { 5 } , \frac { \pi } { 5 } \right)$
B) $\left( - \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$
C) $\left( \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$
D) $\left( - \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$
E) $\left( \frac { \pi } { 3 } , - \frac { \pi } { 3 } \right)$

Question 48

Let $f ( x , y ) = \sin ( x + y ) + \sin x + \sin y$ . Then f has a relative minimum at

A) $\left( - \frac { 5 \pi } { 3 } , \frac { 5 \pi } { 3 } \right)$
B) $\left( \frac { 5 \pi } { 3 } , \frac { 5 \pi } { 3 } \right)$
C) $\left( \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$
D) $\left( - \frac { \pi } { 3 } , \frac { \pi } { 3 } \right)$
E) $\left( \frac { 5 \pi } { 3 } , - \frac { 5 \pi } { 3 } \right)$

Question 49

Let $f ( x , y ) = x ^ { 2 } + y ^ { 3 }$ . Then f has a relative minimum at

A) $( 2,3 )$
B) $( - 2,3 )$
C) $( 0,0 )$
D) $( - 2 , - 3 )$
E) $( 2 , - 3 )$

Question 50

Let $f ( x , y ) = x ^ { 3 } + y ^ { 3 } - 18 x y$ . Then f has a relative minimum at

A) $( 6 , - 6 )$
B) $( 6,6 )$
C) $( 4,2 )$
D) $( - 6,6 )$
E) $( - 6 , - 6 )$

Question 51

Let $f ( x , y ) = \frac { 2 x + 2 y + 1 } { x ^ { 2 } + y ^ { 2 } + 1 }$ . Then f has a relative maximum at

A) $\left( \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$
B) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
C) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 2 } \right)$
D) $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$
E) $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

Question 52

Let $f ( x , y ) = \frac { 2 x + 2 y + 1 } { x ^ { 2 } + y ^ { 2 } + 1 }$ . Then f has a relative minimum at

A) $( - 1 , - 1 )$
B) $( - 1,1 )$
C) $( 1 , - 1 )$
D) $( 1,1 )$
E) $\left( - 1 , - \frac { 1 } { 2 } \right)$

Question 53

Let $f ( x , y ) = 6 x - 24 y - x ^ { 2 } + 2 y ^ { 3 }$ . Then f has a relative maximum at

A) $( 3 , - 1 )$
B) $( 3,1 )$
C) $( 3 , - 2 )$
D) $( 3,2 )$
E) $( - 3,2 )$

Question 54

Let $f ( x , y ) = 2 x ^ { 4 } + y ^ { 2 } - x ^ { 2 } - 2 y$ . Then, the set of saddle points of f is

A) $\{ ( 0,2 , - 1 ) \}$
B) $\{ ( 0,1,1 ) \}$
C) $\{ ( 0,1 , - 1 ) \}$
D) $\{ ( 0 , - 1,1 ) \}$
E) $\{ ( 0 , - 1 , - 1 ) \}$

Question 55

The absolute maximum of $f ( x , y ) = 88 x + 114 y - 2 x ^ { 2 } - 3 y ^ { 2 } - 4 x y$ on the set $\left\{ ( x , y ) : 0 \leq x \leq 50,0 \leq y \leq \frac { 125 } { 3 } \right\}$ is

A)1137
B)1067
C)989
D)923
E)891

Question 56

The absolute maximum of $f ( x , y ) = 2 x + 2 y - x ^ { 2 } - y ^ { 2 } + 2$ on or inside the triangle with vertices (0, 0), (9, 0), and (0,9) is

A)10
B)8
C)6
D)4
E)2

Question 57

The absolute minimum of $f ( x , y ) = 2 x + 2 y - x ^ { 2 } - y ^ { 2 } + 2$ on or inside the triangle with vertices (0, 0), (9, 0), and (0,9) is

A)-69
B)-61
C)-56
D)-51
E)-47

Question 58

The absolute maximum of $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1$ on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

A)5
B)4
C)3
D)2
E)1

Question 59

The absolute minimum of $f ( x , y ) = 2 x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 1$ on or inside the triangle with vertices (0,0), (1,2), and (0,2) is

A)1137
B)-61
C)-5
D)4
E)1

Question 60

The absolute maximum of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } + x y - 6 x$ on or inside the rectangle with vertices (0, -3), (5, -3), (5, 3), and (0, 3) is

A)19
B)16
C)14
D)12
E)8

Question 61

Let $f ( x , y ) = x ^ { 2 } + y$ with constraint $x ^ { 2 } + y ^ { 2 } = 9$ . Then f has a relative minimum at

A) $( - 3,0 )$
B) $( 0,3 )$
C) $( 0 , - 3 )$
D) $( 3,0 )$
E) $( 2 , \sqrt { 5 } )$

Question 62

Let $f ( x , y ) = 5 x - 3 y$ with constraint $x ^ { 2 } + y ^ { 2 } = 136$ . Then f has a relative minimum at

A) $( 6,10 )$
B) $( 6 , - 10 )$
C) $( - 10,6 )$
D) $( 10,6 )$
E) $( 10 , - 6 )$

Question 63

Let $f ( x , y ) = x y$ with constraint $3 x ^ { 2 } + y ^ { 2 } = 6$ . Then f has a relative minimum at

A) $( 1 , \sqrt { 3 } )$
B) $( - 1 , \sqrt { 3 } )$
C) $( 1 , - \sqrt { 3 } )$
D) $( \pm 1 , \pm \sqrt { 3 } ) ( \pm 1 , \mp \sqrt { 3 } )$
E) $( \pm 1 , \mp \sqrt { 3 } )$

Question 64

Let $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ with constraint $3 x + y = 3$ . Then f has a relative minimum at

A) $\left( \frac { 9 } { 10 } , \frac { 3 } { 10 } \right)$
B) $\left( - \frac { 9 } { 10 } , \frac { 3 } { 10 } \right)$
C) $\left( \frac { 9 } { 10 } , - \frac { 3 } { 10 } \right)$
D) $\left( \frac { 3 } { 10 } , \frac { 9 } { 10 } \right)$
E) $\left( \frac { 3 } { 10 } , - \frac { 9 } { 10 } \right)$

Question 65

Let $f ( x , y , z ) = x y z$ with constraint $x + y + z = 1$ . Assume that $x , y , z \geq 0$ . Then f has a relative maximum at

A) $\left( \frac { 1 } { 3 } , \frac { 1 } { 6 } , \frac { 1 } { 2 } \right)$
B) $\left( \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 1 } { 6 } \right)$
C) $\left( \frac { 1 } { 2 } , \frac { 1 } { 6 } , \frac { 1 } { 3 } \right)$
D) $\left( \frac { 1 } { 3 } , \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
E) $\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 6 } \right)$

Question 66

Let $f ( x , y , z ) = x ^ { 2 } - y ^ { 2 }$ with constraint $x ^ { 2 } + 2 y ^ { 2 } + 3 z ^ { 2 } = 1$ . Then f has a relative minimum at

A) $( \pm 1,0,0 )$
B) $( - 1,0,0 )$
C) $( 1,0,0 )$
D) $\left( \frac { 1 } { 2 } , 0 , \frac { 1 } { 2 } \right)$
E) $\left( \frac { 1 } { 2 } , 0 , - \frac { 1 } { 2 } \right)$

Question 67

Let $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

A) $( 1 , \pm 1 , \pm 1 )$
B) $( - 1 , \pm 1 , \mp 1 )$
C) $( 1,1,1 ) , ( - 1 , - 1 , - 1 )$
D) $( 1 , - 1 , - 1 ) , ( - 1,1 , - 1 )$
E) $( 1 , \pm 1 , \pm 1 ) , ( - 1 , \pm 1 , \mp 1 )$

Question 68

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative maximum at

A) $( \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$
B) $( \sqrt { 3 } , - \sqrt { 3 } , \sqrt { 3 } )$
C) $( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$
D) $( - \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$
E) $( - \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$

Question 69

Let $f ( x , y , z ) = x + y + z$ with constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Then f has a relative minimum at

A) $( \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$
B) $( - \sqrt { 3 } , - \sqrt { 3 } , - \sqrt { 3 } )$
C) $( \sqrt { 3 } , \sqrt { 3 } , \sqrt { 3 } )$
D) $( - \sqrt { 3 } , \sqrt { 3 } , - \sqrt { 3 } )$
E) $( \sqrt { 3 } , - \sqrt { 3 } , - \sqrt { 3 } )$

Question 70

Let $f ( x , y , z ) = x y z$ with constraint $2 x y + 3 x z + y z = 72$ . Then f has a relative maximum at

A) $( - 2 , - 6 , - 4 )$
B) $( 2 , - 6 , - 4 )$
C) $( - 2 , - 6,4 )$
D) $( 2,6,4 )$
E) $( - 2,6 , - 4 )$

Question 71

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint xyz = 1. Then f has a relative minimum at

A) $\left( 2 , \pm 1 , \pm \frac { 1 } { 2 } \right) , \left( - 2 , - 1 , \frac { 1 } { 2 } \right)$
B) $\left( 2 , \pm 1 , \pm \frac { 1 } { 2 } \right) , \left( - 2,1 , - \frac { 1 } { 2 } \right)$
C) $\left( 2 , - 1 , - \frac { 1 } { 2 } \right) , \left( - 2 , \pm 1 , \pm \frac { 1 } { 2 } \right)$
D) $\left( 2,1 , \frac { 1 } { 2 } \right) , \left( - 2 , \pm 1 , \pm \frac { 1 } { 2 } \right)$
E) $\left( 2 , \pm 1 , \pm \frac { 1 } { 2 } \right) , \left( - 2 , \pm 1 , \pm \frac { 1 } { 2 } \right)$

Question 72

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint xy = 1. Then f has a relative minimum at

A) $\left( \sqrt { 2 } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$
B) $\left( - \sqrt { 2 } , - \frac { 1 } { \sqrt { 2 } } , 0 \right)$
C) $\left( \frac { 1 } { \sqrt { 2 } } , \sqrt { 2 } , 0 \right)$
D) $\left( \pm \sqrt { 2 } , \pm \frac { 1 } { \sqrt { 2 } } , 0 \right)$
E) $\left( - \frac { 1 } { \sqrt { 2 } } , - \sqrt { 2 } , 0 \right)$

Question 73

Let $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + 16 z ^ { 2 }$ with constraint x = 1. Then f has a relative minimum at

A) $\left( 1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$
B) $\left( 1 , \frac { 1 } { 2 } , - \frac { 1 } { 4 } \right)$
C) $( 1,0,1 )$
D) $\left( 1 , - \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$
E) $( 1,0,0 )$

Question 74

Let $w = ( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + z ^ { 2 }$ with constraint $4 x + 2 y - z = 5$ . Then the minimum value of w is

A) $\frac { 29 } { 21 }$
B) $\frac { 25 } { 21 }$
C) $\frac { 23 } { 21 }$
D) $\frac { 20 } { 21 }$
E) $\frac { 19 } { 21 }$

Question 75

Let $w = ( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + ( z + 1 ) ^ { 2 }$ with constraint $x + 4 y + 3 z = 2$ . Then the minimum value of w is

A) $\frac { 37 } { 13 }$
B) $\frac { 35 } { 13 }$
C) $\frac { 33 } { 13 }$
D) $\frac { 32 } { 13 }$
E) $\frac { 30 } { 13 }$

Question 76

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $x ^ { 2 } + 4 y ^ { 2 } = 16$ . Then the minimum value of w is

A)24
B)23
C)19
D)17
E)16

Question 77

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $x + y + z = 1$ . Then the maximum value of w is

A) $\frac { 1 } { 3 }$
B)1
C) $\frac { 4 } { 3 }$
D) $\frac { 5 } { 4 }$
E)2

Question 78

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $9 x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } = 36$ . Then the minimum value of w is

A)4
B)2
C)1
D) $\frac { 2 } { 3 }$
E) $\frac { 1 } { 3 }$

Question 79

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $9 x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 } = 36$ . Then the maximum value of w is

A)48
B)40
C)36
D)32
E)28

Question 80

Let $w = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ with constraint $3 x - y + 5 z = 1$ . Then the minimum value of w is

A) $\frac { 1 } { 13 }$
B) $\frac { 1 } { 19 }$
C) $\frac { 1 } { 21 }$
D) $\frac { 1 } { 35 }$
E) $\frac { 1 } { 41 }$