(i) Maximize subject to the constraint xS1U1B11S1U1B0S1U1P12S1S1P0 +xS1U1B12S1U1B0S1U1P12S1S1P0 + ..... xS1U1B1nS1U1B0S1U1P12S1S1P0 = 1. (ii) Use part (i) to prove the well-known Arithmetic-Geometric Inequality : For any positive real numbers y1 , y2 , ....... yn ,

The answer of (i) Maximize subject to the...

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 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

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

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

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) 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)

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. 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.

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

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

Exam 14: Applications of Partial Derivatives

(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 ,

(Essay)

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Question 41

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.

(Multiple Choice)

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Question 42

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.

(Multiple Choice)

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Question 43

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

(Multiple Choice)

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Question 44

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 = $\pi$ .

(Multiple Choice)

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Question 45

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.

(Multiple Choice)

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Question 46

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

(Multiple Choice)

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Question 47

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

(Multiple Choice)

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Question 48

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

(Multiple Choice)

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Question 49

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

(Multiple Choice)

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Question 50

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

(Multiple Choice)

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Question 51

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.

(Multiple Choice)

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Question 52

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.

(Essay)

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Question 53

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

(Multiple Choice)

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Question 54

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

(Essay)

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Question 55

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

(Multiple Choice)

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Question 56

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.

(Multiple Choice)

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Question 57

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

(Multiple Choice)

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Question 58

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.

(Multiple Choice)

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Question 59

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

(Essay)

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Question 60

Showing 41 - 60 of 67

(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 ,

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.

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.

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

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 = $\pi$ .

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.

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

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

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

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

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

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.

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

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

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

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.

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

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.

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

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

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Exam 14: Applications of Partial Derivatives

(i) Maximize subject to the constraint x12 +x22 + ..... xn2 = 1. (ii) Use part (i) to prove the well-known Arithmetic-Geometric Inequality : For any positive real numbers y1 , y2 , ....... yn ,

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.

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.

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

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 = π\piπ , and y = π\piπ .

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.

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

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

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

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

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

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.

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

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

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

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.

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

Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y) = x2y + z subject to the constraints x2 + y2 = 1 and z = y.

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

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

(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 ,

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.

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.

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

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 = $\pi$ .

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.

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

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

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

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.

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

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.

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

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.