1. Find all the local maxima, local minima, and saddle points of the function.
2
2
f(x,y) = x + xy + y + 5x − 5y + 4
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. A local maximum occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
The local maximum value(s) is/are .
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local maxima.
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. A local minimum occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
The local minimum value(s) is/are .
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local minima.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. A saddle point occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There are no saddle points.
Answers B. There are no local maxima.
A. A local minimum occurs at ( − 5,5)
.
(Type an ordered pair. Use a comma to separate answers as needed.)
The local minimum value(s) is/are − 21
.
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no saddle points.
ID: 14.7.1
2. Find all the local maxima, local minima, and saddle points of the function shown below.
f(x,y) = 2 −
5
2
x +y
2
Find the local maxima. Select the correct choice below and, if necessary, fill in the answer boxes as needed to complete your choice.
A. A local maximum occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
The local maximum value(s) is/are .
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local maxima.
Find the local minima. Select the correct choice below and, if necessary, fill in the answer boxes as needed to complete your choice.
A. A local minimum occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
The local minimum value(s) is/are .
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local minima.
Find the saddle points. Select the correct choice below and, if necessary, fill in the answer box as needed to complete your choice.
A. A saddle point occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There are no saddle points.
Answers A. A local maximum occurs at (0,0)
.
(Type an ordered pair. Use a comma to separate answers as needed.)
The local maximum value(s) is/are 2
.
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local minima.
B. There are no saddle points.
ID: 14.7.12
3. Find all the local maxima, local minima, and saddle points of the function.
f(x,y) =
e
−y
2
2
(x + y ) + 1
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. A local maximum occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
The local maximum value(s) is/are .
(Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local maxima.
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. A local minimum occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
The local minimum value(s) is/are .
(Type anexact answer. Use a comma to separate answers as needed.)
B. There are no local minima.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. A saddle point occurs at .
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There are no saddle points.
Answers B. There are no local maxima.
A. A local minimum occurs at (0,0)
.
(Type an ordered pair. Use a comma to separate answers as needed.)
The local minimum value(s) is/are 1
.
(Type anexact answer. Use a comma to separate answers as needed.)
A. A saddle point occurs at (0,2)
.
(Type an ordered pair. Use a comma to separate answers as needed.)
ID: 14.7.27
4. Find the absolute maximum and minimum of the function on the given domain.
2
2
f(x,y) = 4x + 5y on the closed triangular plate bounded by the lines x = 0, y = 0, y + 2x = 2 in the first quadrant
The absolute maximum is (Simplify your answer.)
.
The absolute minimum is (Simplify your answer.)
.
Answers 20
0
ID: 14.7.33
5. Find the absolute maximum and minimum of the function f(x,y) = (48x − 12x2 ) cos y on the π
rectangular plate 1 ≤ x ≤ 3, −
4
π
≤y≤
4
.
The absolute maximum is .
The absolute minimum is .
Answers 48
18 2
ID: 14.7.37
6. Find the point on the plane 4x + 3y + z = 10 that is nearest the origin.
What are the values of x, y, and z for the point?
x=
y =
z =
(Type integers or simplified fractions.)
Answers 20
13
15
13
5
13
ID: 14.7.51
7. Find three numbers whose sum is 30 and whose sum of squares is a minimum.
The three numbers are .
(Use a comma to separate answers as needed.)
Answer: 10,10,10
ID: 14.7.53
2
z= (48x− 12x ) cos y
8. Find the extreme values of f(x,y) = xy subject to the constraint x2 + y2 − 24 = 0.
The extrema are .
(Use a comma to separate answers as needed. Simplify your answer.)
Answer: 12, − 12
ID: 14.8.2
9. Use the method of Lagrange multipliers to find
a. the minimum value of x + y, subject to the constraints xy = 4, x > 0, y > 0.
b. the maximum value of xy, subject to the constraint x + y, = 4.
The minimum value of x + y is (Simplify your answer.)
The maximum value of xy is (Simplify your answer.)
.
.
Answers 4
4
ID: 14.8.7
10. The temperature at point (x,y) on a metal plate is T(x,y) = 256x2 − 256xy + 64y2 . An ant on the plate walks around the circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant?
The minimum temperature is degrees.
The maximum temperature is degrees.
Answers 0
8000
ID: 14.8.15
11. Find the point on the sphere x2 + y2 + z2 = 16 farthest from the point (1, − 1, − 1).
The point is (
Answers
,
,
). (Type an ordered triple.)
4
−
3
4
3
4
3
ID: 14.8.18
12. Find three real numbers x, y, and z whose sum is 12 and the sum of whose squares is as small as possible.
The three numbers are .
(Simplify your answers. Use a comma to separate answers as needed.)
Answer: 4,4,4
ID: 14.8.25
13. Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere.
The dimensions are .
(Type exact answers, using radicals as needed. Use a comma to separate answers as needed.)
Answer: 2 3 2 3 2 3
,
,
3
3
3
ID: 14.8.27
14. A space probe in the shape of the ellipsoid 36x2 + y2 + 4z2 = 41 enters a planet's atmosphere and its surface begins to 2
heat. After 1 hour, the temperature at the point (x,y,z) on the probe's surface is T(x,y,z) = 72x + 4yz − 16z + 596. Find the hottest point on the probe's surface.
The hottest point is ( ±
,
,
).
(Simplify your answer. Type exact answers, using radicals as needed. Use integers or fractions for any numbers in the
expression.)
Answers 17
18
−
−
ID: 14.8.29
4
3
4
3