Deck 14: Applications of Partial Derivatives

Question 1

Find the critical points of f(x, y) = 3x² - 2xy + 2y² - 10x + 1.

A) (2, 1)
B) (1, 1)
C) (0, 0)
D) (1, -1) and (-1, 1)
E) (2, 1) and (1, 2)

Answer 1

(2, 1)

Question 2

Find the critical points of f(x, y) = ln (x² + y² + 4x - 4y + 8).

A) (-2, 2)
B) (2, -2)
C) (-2, 2) and (2, -2)
D) (-2, 2), (2, -2), and (0, 0)
E) There are no critical points.

Answer 2

There are no critical points.

Question 3

Find all critical points of f(x) = 2x³y -4x³ + 6y³ -18y + 19.

A) (0, 1), (0, - 1), and (-9, 2)
B) only (-3, 2)
C) only (3, 2)
D) (0, 1), (0, -1), and (-3, 2)
E) only (0, 1) and (-3, 2)

Answer 3

(0, 1), (0, -1), and (-3, 2)

Question 4

Find and classify the critical points of the function f(x, y) = x sin y.

A) local maxima at (0, n $\pi$ ) for even integers n, local minimum at (0, n $\pi$ ) for odd integers n
B) local minima at (0, n $\pi$ ) for even integers n, local maximum at (0, n $\pi$ ) for odd integers n
C) saddle points at all points (0, n $\pi$ ) for integers n
D) only critical point is (0, 0), a saddle point
E) local minima at (0, n $\pi$ ) for nonzero integers n, saddle point at (0, 0)

Question 5

Find and classify the critical points of the function f(x, y) = 2y³ - 3x² - 3xy + 9x.

A) local minimum at (2, -1), saddle point at
B) local maximum at (2, -1), saddle point at
C) saddle point at (2, -1), local maximum at
D) saddle point at (2, -1), local minimum at
E) saddle point at (2, -1) and

Question 6

Find and classify all critical points for the function f(x, y) = .

A) (1, 0); minimum
B) (-1, 0); maximum
C) (0, 0); minimum
D) (0, 0); saddle
E) There are no critical points.

Question 7

Find and classify all critical points for the function f(x, y) = x³ - 12xy² + y³ + 45y.

A) (2, 1) and (-2, -1) are saddle points
B) (2, 1), (2, -1), (-2, 1), and (-2, -1) are saddle points
C) (2, 1) is a local maximum and (-2, -1) is a local minimum
D) (2, -1) and (-2, 1) are saddle points
E) There are no critical points.

Question 8

Find the maximum and minimum values of f(x, y) = (x - y) .

A) maximum , minimum 0
B) maximum , minimum
C) maximum , minimum -
D) maximum 2 , minimum 0
E) maximum , minimum -

Question 9

Find and classify all critical points for the function f(x, y) = 2y³ + 3y² - 12y -x² + 2x.

A) (1, -2) local maximum; (1, 1) local minimum
B) (1, -2) local maximum; (1, 1) saddle point
C) (1, -2) local maximum; (1, 1) local maximum
D) (1, -2) local minimum; (1, 1) local minimum
E) (-1, 2) saddle point; (1, 1) saddle point

Question 10

A rectangular box with no top has given surface area S. Find the length, width, and height of the box if the volume is as large as possible.

A) length = width = , height =
B) length = width = , height =
C) length = width = , height =
D) length = width = , height =
E) length = width = , height =

Question 11

Find and classify the critical points of the following function: f(x, y) = + 30x³ - 15y³.

A) saddle points are ( , - ) and (- , ), maximum at (3, 3) and minimum at (-3, -3)
B) saddle points are (0, 0), ( , - ) and (- , ), minimum at (3, 3) and maximum at (-3, -3)
C) saddle points are (0, 0), ( , - ), (- , ), (3, 3), and (-3, -3)
D) saddle points are (0, 0), ( , - ) and (- , ), maximum at (3, 3) and minimum at (-3, -3)
E) saddle points are (0, 0), ( , - ) and (- , ), maximum at (3, -3) and minimum at (-3, 3)

Question 12

Find and classify all critical points of f(x,y,z) = x³ + xz² + 3x² + y² + 2z² - 9x - 2y -10.

A) local minimum at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0) and saddle point at (1, 1, 0)
B) local minimum at (-2, 1, 3), (-2, 1, -3) and local maximum at (-3, 1, 0), (1, 1, 0)
C) local minimum at (1, 1, 0 ) and saddle point at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0)
D) local maximum at (-2, 1, 3), (-2, 1, -3), (-3, 1, 0) and saddle point at (1, 1, 0)
E) local minimum at (-2, 1, 3) , (-3, 1, 0), local maximum at (1, 1, 0), and saddle point at (-2, 1, -3)

Question 13

If a function f(x,y) has a local or absolute extreme value at the point (x₀, y₀) in its domain, then (x₀, y₀) must be either a critical point of f, a singular point of f, or a boundary point of the domain of f.

Question 14

Suppose the critical points of f(x,y) = 2x³ - 15x² + 24x - y² - 6y + 1 occur at the points(1, -3) and (4, -3).Then the function f has:

A) saddle points at (1, -3) and (4, -3)
B) a local minimum at (4, - 3) and a saddle point at (1, -3)
C) a local maximum at (1, - 3) and a saddle point at (4, - 3)
D) a local maximum at (4, -3) and a saddle point at (1, -3)
E) a local maximum at (4, -3) and a local minimum at (1, -3)

Question 15

Suppose that a function f(x,y) has a critical point (a, b) at an interior point in its domain and that f has continuous second order partials in a neighbourhood of (a, b).
If , then f has no local extremum at (a, b).

Question 16

A closed rectangular container of volume 96 cubic metres is to be made from three different materials.The top and the bottom of the container are to be made from a material that costs $4 per square metre, two parallel sides (say left and right) are to be made from a material that costs $3 per square metre, and the other two parallel sides (front and back) are to be made from a material that costs $1 per square metre.Let x and y be the dimensions of the base of the container and z be its height in metres.(i) Express the total cost of the container (in dollars) as a function of x and y.(ii) Find dimensions of the most economical container and how much it costs.

Question 17

Find the absolute maximum and minimum values of f(x, y) = y² - y - x - 5 on the square0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1.

A) maximum 0, minimum -
B) maximum - , minimum -
C) maximum -5, minimum -
D) maximum -5, minimum -
E) maximum - , minimum -5

Question 18

Find the absolute maximum and minimum of f(x, y) = 4x² + 2xy - 3y² on the unit square0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1.

A) maximum , minimum -3
B) maximum 4, minimum -3
C) maximum 4, minimum 0
D) maximum , minimum -4
E) maximum , minimum 0

Question 19

Find the absolute maximum and minimum values of f(x, y) = 4(x - x²) sin( $\pi$ y) on the rectangle 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 2 and the points where they are assumed.

A) maximum 1 at , minimum 0 at
B) maximum 2 at , minimum -2 at
C) maximum 1 at , minimum -1 at
D) maximum 1 at , minimum 0 at (0, 0), (1, 0), (0, 2), and (1, 2)
E) maximum 1 at , minimum 0 at

Question 20

Find the absolute maximum and minimum values of the linear function f(x, y) = -2x + y - 10 on the polygon 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 2, y - x $\le$ 1.

A) maximum -9, minimum -14
B) maximum -9, minimum -12
C) maximum -8, minimum -12
D) maximum -8, minimum -15
E) maximum -9, minimum -10

Question 21

Find the absolute maximum and minimum values of f(x, y) = x² - 3x + y² - 3y + 5 on the triangle bounded by x = 0, y = 0, and x + y = 2.

A) maximum 5, minimum 1
B) maximum 5, minimum 0
C) maximum 6, minimum -1
D) maximum 6, minimum 0
E) maximum 5, minimum

Question 22

Find the absolute maximum and minimum values of f(x, y, z) = xyz(1 - x² - y² - z²) on the tetrahedron bounded by x = 0, y = 0, z = 0, and x + y + z = 2.

A) maximum , minimum 0
B) maximum , minimum 0
C) maximum , minimum 0
D) maximum , minimum 0
E) none of the above

Question 23

Find the absolute maximum and minimum values of f(x, y) = xy on the disk x² + y² $\le$ 1.

A) maximum 1, minimum -1
B) maximum , minimum -
C) maximum , minimum -
D) maximum , minimum 0
E) none of the above

Question 24

Find the absolute maximum and minimum values of f(x, y) = 2 - x² - 4y on the diskx² + y² $\le$ 9.

A) maximum 16, minimum -14
B) maximum 10, minimum -13
C) maximum 14, minimum -11
D) maximum 12, minimum -10
E) none of the above

Question 25

Find the absolute maximum and minimum values of on the semi-infinite strip 0 ≤ x < ∞, -1 ≤ y ≤ 2, if these extreme values exist.

A) maximum , minimum -
B) maximum , minimum -
C) maximum , minimum -
D) maximum , minimum 0
E) no extreme values exist

Question 26

Find the absolute maximum and absolute minimum values of f(x , y) = cos(x) + cos(y) - cos(x + y) - 1 on the closed square region bounded by the straight lines x = 0, y = 0, x = $\pi$ , and y = $\pi$ .

A) absolute minimum 0, absolute maximum
B) absolute minimum -4, absolute maximum - 1
C) absolute minimum -4, absolute maximum
D) absolute minimum -2, absolute maximum 0
E) absolute minimum -4, no absolute maximum

Question 27

Use Lagrange multipliers to find the extreme values of f(x, y) = x² + 3y² + 2y on the unit circle x² + y² = 1.

A) maximum 6, minimum -
B) maximum 5, minimum -
C) maximum 6, minimum 0
D) maximum 5, minimum
E) maximum 5, minimum 1

Question 28

Find the maximum and minimum values of f(x, y) = x² + 3y² + 2y on the disk x² + y² $\le$ 1. Use Lagrange multipliers to handle the boundary analysis.

A) maximum 6, minimum -
B) maximum 5, minimum -
C) maximum 6, minimum 0
D) maximum 5, minimum
E) maximum 5, minimum

Question 29

Find the maximum and minimum distances from the origin to the ellipse 5x² + 6xy + 5y² - 8 = 0.

A) maximum 2, minimum 1
B) maximum 4, minimum 1
C) maximum 2, minimum
D) maximum 4, minimum
E) maximum 4, minimum 2

Question 30

Find the point on the sphere x² + y² + z² = 10 that is closest to the point (1, -8, 5).

A)
B)
C) (0, , 0)
D)
E) (2, -16, 10)

Question 31

Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy²z³ on the sphere x² + y² + z² = 6.

A) maximum 6, minimum -6
B) maximum 6 , minimum -6
C) maximum 6 , minimum -6
D) maximum 12, minimum -12
E) maximum 12 , minimum -12

Question 32

Find the point on the surface z = x² + y² closest to the point (1, 1, 0).

A) (1, 1, 1)
B)
C)
D)
E)

Question 33

Find the maximum and minimum values of f(x , y) = x² + xy + 2y² subject to the constraint x² + 3y² = 3.

A) minimum 0 , maximum 3
B) minimum 2 , maximum 3
C) minimum , maximum
D) minimum 3 , maximum
E) minimum , maximum 2

Question 34

Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y) = x^2y + z subject to the constraints x² + y² = 1 and z = y.

A) ±
B) ±
C) ±
D) ±
E) ±

Question 35

Let f(x, y, z) = x² + 2y² + 4z². Find the point on the plane x + y + z = 14 at which f has its smallest value.

A) (8, 4, 2)
B) (0, 0, 14)
C) (1, 2, 11)
D)
E) (2, 4, 8)

Question 36

Find the points closest to the origin on the hyperbola in which the cone x² + y² = z² intersects the plane x + y = 2.

A) (1, 1, ± )
B) (-1, -1, ± )
C) (-1, -1, ± ) and (1, 1, ± )
D) (2, 0, 2) and (0, 2, 2)
E) (1, 1, ) and (-1, -1, - )

Question 37

Find the maximum and minimum values of the function f(x, y, z) = 3x² + 3y² + 5 + 2xy - 2xz - 2yz over the sphere x² + y² + z² = 6.

A) maximum value 18, minimum value 12
B) maximum value 36, minimum value 12
C) maximum value 20, minimum value 18
D) maximum value 36, minimum value 18
E) maximum value 24, minimum value 16

Question 38

If the Lagrange function L corresponding to the problem of extremizing f(x, y, z) subject to the constraint g(x, y, z) = 0 has exactly two critical points, then f must attain its maximum value at one of the points and attain its minimum value at the other point.

Question 39

The extreme values of the function f(x , y, z) = 23 x + y²z subject to the constraintsx - z = 0 and y² + z² = 36 are given by:

A) 115, 128
B) - 115, 115
C) -128, 115
D) -128, 128
E) -128, -115

Question 40

Find the maximum and minimum values of the function f(x, y, z, u, v) = x² + y² + z² + u² + v² subject to the constraints x + y + 3z = 7 and 3z - u -v = 13.

A) maximum 19, occurs at the point (x, y, z, u, v) = (- 1, -1, 3, -2, -2) and no minimum value
B) minimum 19, occurs at the point (x, y, z, u, v) = (- 1,- 1, 3, -2, -2) and no maximum value
C) maximum 89, occurs at the point (x, y, z, u, v) = (1, 1, -9, 2, 2) and no minimum value
D) minimum 0, occurs at the point (x, y, z, u, v) = (0, 0, 0, 0, 0) and no maximum value
E) There are no finite extreme values.

Question 41

Let pi > 0, i = 1, 2, 3,..., n be real numbers such that
Find the maximum value of subject to the constraint

Question 42

Use Lagrange multipliers to find the point P( , , ) on the sphere + + = 9 and the point Q ( , , ) on the plane x + 2y - 2z = 7 where the distance between P and Q is minimum and determine this minimum distance.

A) P (2, -4, 7), Q (5, 2, 1); minimum distance 9 units
B) P (4, 0, 3), Q (3, -2, 5); minimum distance 3 units
C) P (4, 0, 3), Q (5, 2, 1); minimum distance 3 units
D) P (2, -4, 7), Q (-1, -10, 13); minimum distance 9 units
E) P (2, -4, 7), Q (1, -6, 9); minimum distance 3 units

Question 43

Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = .

Question 44

(i) Maximize subject to the constraint x₁² +x₂² + ..... x_n² = 1.
(ii) Use part (i) to prove the well-known Arithmetic-Geometric Inequality :

For any positive real numbers y1 , y2 , ....... yn ,

Question 45

Starfleet Command has given Captain Kirk a subspace amplifier to channel communications among n star systems having galactic Cartesian coordinates ( , , ) for k = 1, 2,..., n. To what point in space should Captain Kirk take the Enterprise to place the amplifier so as to minimize the sum of the squares of its distances from the n star systems?

A) the point
B) the point
C) the point
D) the point
E) the point

Question 46

Find an equation of the line of best fit given the following points: (1, -1), (2, 1), (3, 2), (4, 1), and (5, 0).

A) y =
B) y = 5x
C) y = 2x -1
D) y = + 1
E) y = - 1

Question 47

Find the value of the constant m so that the line y = mx best fits the experimental data (1, 2.3), (2, 4.2), (3, 7.1), (4, 8.8) in the sense of minimizing the sum of the squares of the vertical distances of the data points from the line.

A) m = 2.36
B) m = 2.24
C) m = 2.18
D) m = 2.11
E) m = 2.34

Question 48

Find the value of the constant a so that the graph of the function f(x) = ax² best fits the curve y = on the interval [0, 1] in the sense of minimizing the integral of the square of the vertical distance between the two curves on that interval.

A) a =
B) a =
C) a =
D) a =
E) a =

Question 49

Find the empirical regression line y = a + bx for the data (x , y) = (1, -2), 4, 1), and (1, 10). What is the predicted value of y when x = -3?

Question 50

Find the values of constant real numbers a and b to minimize dx.

A) a = 15 , b = -1
B) a = 19 , b = -
C) a = - 24 , b = 4
D) a = - , b =
E) a = 24 , b = - 4

Question 51

Find the derivative of the function f(x) = dt.

A) 6x
B) 4x
C) 2x
D) 2x
E) 2x

Question 52

Use the fact that = ln (1 + ab) to evaluate dx.

A) ln (1 + ab) +
B) ln (1 + ab) +
C) ln (1 + ab) -
D) ln (1 + ab) -
E) ln (1 + ab) +

Question 53

By first differentiating the integral, evaluate dy for x > -1.

A) ln (2x + 1)
B) ln (|x|) + 1
C) ln (x - 1)
D) ln (x + 1)
E) ln (x) + 1

Question 54

Solve the integral equation f(x) = 3 + 2 dt.

A) f(x) = 2
B) f(x) = 3
C) f(x) = 2
D) f(x) = 3
E) f(x) = 3

Question 55

Find the envelope of the family of straight lines y - 3 x + 2 = 0.

A) y = x³
B) y = 2x³
C) y² = x³
D) y² = 3x³
E) y = 3x³

Question 56

Given F( ) = = , for > .
By first differentiating F with respect to 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11, evaluate .

A)
B)
C) -
D)
E)

Question 57

Evaluate F(x,y) = dt for x > 0, y > 0.

Question 58

Find the envelope of the family of straight lines xcosh(c) + ysinh(c) = 3.

Question 59

Every one-parameter family of curves in the plane has an envelope.

Question 60

Use Newton's method to solve the system ( + y) = 2, - = 1.

A) x = 1.0502656, y = 0.54118591
B) x = 1.0503479, y = 0.54110211
C) x = 1.0510302, y = 0.54116463
D) x = 1.0514452, y = 0.54230058
E) x = 1.0523415, y = 0.54435746

Question 61

Use Newton's method to find a first quadrant solution of the system x² + y⁴ = 1, y³ = ( ).

A) x = 0.40784120, y = 0.96737370
B) x = 0.88321233, y = 0.76223749
C) x = 0.72413552, y = 0.74316200
D) x = 0.76006565, y = 0.80613044
E) x = 0.64874536, y = 0.64385647

Question 62

Use Maple's fsolve with Digits : = 6 to find a solution to the system + = 3, x sin(x) - y cos(y) = 0 near (1, 2).

A) x = 0.461289, y = 1.42635
B) x = 0.462289, y = 1.53635
C) x = 0.465489, y = 1.48635
D) x = 0.460089, y = 1.42335
E) x = 0.463445, y = 1.41657

Question 63

Find and classify the critical points of f(x, y) = (2 - xy + 5 - x + 2y) .

Question 64

Find the maximum and minimum values of the function f(x, y) = .

A) maximum $~~~ ~$ 0.145490, $~~~ ~$ minimum - 0.158939
B) maximum $~~~ ~$ 0.146110, $~~~ ~$ minimum - 0.160236
C) maximum $~~~ ~$ 0.141191, $~~~ ~$ minimum - 0.152573
D) maximum $~~~ ~$ 0.129903, $~~~ ~$ minimum - 0.173412
E) maximum $~~~ ~$ 0.145276, $~~~ ~$ minimum - 0.168873

Question 65

Find the minimum value of the function f(x, y) = x⁴ + y⁴ - xy + xy²

A) -0.382006
B) -0.362705
C) -0.077091
D) -0.051392
E) -0.243454

Question 66

Use Maple's fsolve routine to solve the non-linear system of equations Quote the solution to 5 significant figures.

Question 67

Find an optimal binary compression for a 16-character alphabet with probabilities given in the table below. Use four figures of accuracy.

A) 3.0139
B) 0.8398
C) 0.5821
D) 0.2528
E) 0.3607