Optimization Problems: Master Calculus III Concepts with Interactive Quizzes

Question 1

Find the maximum value of the function f(x, y) = xy on the ellipse x²/9 + y²/4 = 1.

Accepted Answer

6

Answer

3/2

Answer

12

Answer

24

Question 2

Solve the following linear programming problem: maximize z = 2x + 3y subject to the constraints x + y ≤ 5, x ≥ 0, y ≥ 0.

Accepted Answer

x = 2.5, y = 2.5

Answer

x = 0, y = 5

Answer

x = 5, y = 0

Answer

x = 0, y = 2.5

Question 3

Find the minimum of the function f(x, y, z) = x² + y² + z² subject to the constraint xy + yz + zx = 1.

Accepted Answer

(1/3, 1/3, 1/3)

Answer

(0, 1, 0)

Answer

(1, 0, 0)

Answer

(1/2, 1/2, 0)

Question 4

Find the maximum value of the function f(x, y, z) = xyz on the sphere x² + y² + z² = 4.

Accepted Answer

16/3

Answer

8/3

Answer

32/3

Answer

64/3

Question 5

Solve the following nonlinear programming problem: minimize f(x, y) = x² + y² subject to the constraint x + y = 4.

Accepted Answer

(2, 2)

Answer

(4, 0)

Answer

(0, 4)

Answer

(1, 3)

Question 6

Find the maximum value of the function f(x, y) = xy, subject to the constraint x + y = 10.

Accepted Answer

25

Answer

10

Answer

20

Answer

50

Question 7

Find the critical points of the function f(x, y) = x^3 + y^3 - 3xy.

Accepted Answer

(0, 0) and (1, 1)

Answer

(0, 0) only

Answer

No critical points

Answer

(1, 1) only

Question 8

Determine the minimum value of the function f(x, y) = x^2 + y^2.

Accepted Answer

0

Answer

-1

Answer

2

Answer

1

Question 9

Minimize the function f(x, y) = x^2 + y^2 subject to the constraint 2x + 3y = 12.

Accepted Answer

9/13

Answer

2

Answer

1/2

Answer

4

Question 10

Find the optimal solution to the following linear programming problem: maximize z = 2x + 3y subject to the constraints x + y ≤ 4, x ≥ 0, and y ≥ 0.

Accepted Answer

(2, 2)

Answer

(4, 0)

Answer

(0, 4)

Question 11

Find the direction in which the function f(x, y) = xy + y^2 increases most rapidly at the point (2, 1).

Accepted Answer

(1, 2)

Answer

(2, 1)

Answer

(-1, -2)

Answer

(0, 0)

Question 12

Find the maximum value of the function $f(x, y) = x + y$ subject to the constraints $x ge 0, y ge 0$, and $x + y le 10$.

Accepted Answer

10

Answer

5

Answer

20

Answer

15

Question 13

What are the necessary conditions for a point $(x, y)$ to be a critical point of a function $f(x, y)$?

Accepted Answer

The partial derivatives of $f$ with respect to $x$ and $y$ are both zero at $(x, y)$.

Answer

The partial derivative of $f$ with respect to either $x$ or $y$ is zero at $(x, y)$.

Answer

The partial derivative of $f$ with respect to $x$ is nonzero at $(x, y)$.

Question 14

Which of the following is NOT a method for solving optimization problems?

Accepted Answer

Numerical integration

Answer

Linear programming

Answer

Graphical methods

Answer

Method of Lagrange multipliers

Question 15

Distinguish between a local maximum and a global maximum.

Accepted Answer

A local maximum is the largest value of the function in a small neighborhood around a point, while a global maximum is the largest value of the function over the entire domain.

Answer

A local maximum is the smallest value of the function in a small neighborhood around a point, while a global maximum is the smallest value of the function over the entire domain.

Question 16

Explain the purpose of the second derivative test.

Accepted Answer

To determine whether a critical point is a local maximum, local minimum, or saddle point.

Answer

To optimize a function subject to constraints.

Answer

To find the critical points of a function.

Question 17

Determine the critical points of the function *f(x, y) = x³ - 3xy + y³*.

Accepted Answer

(0, 0) and (1, 1)

Answer

(0, 0) and (1, -1)

Answer

(-1, -1) and (1, 1)

Answer

(1, 0) and (0, 1)

Question 18

What is the minimum value of the function *f(x, y) = x² + y²* on the circle *x² + y² = 1*?

Accepted Answer

0

Answer

1

Answer

4

Answer

2

Question 19

Identify the saddle point of the function *f(x, y) = x³ - 3xy² + y³*.

Accepted Answer

(0, 0)

Answer

(0, 1)

Answer

(1, 1)

Answer

(1, 0)

Question 20

Determine the maximum and minimum values of the function *f(x, y) = x² - 2y²* on the region defined by *x² + y² ≤ 1*.

Accepted Answer

Maximum: 1, Minimum: -2

Answer

Maximum: 2, Minimum: -2

Answer

Maximum: 2, Minimum: -1

Answer

Maximum: 1, Minimum: -1

Question 21

Find the critical points of the function *f(x,y) = x^2 + y^2 - 4x + 2y* subject to the constraint *x^2 + y^2 = 1*.

Accepted Answer

(1, 0) and (-1, 0)

Answer

(1, 1) and (-1, -1)

Answer

(2, 0) and (-2, 0)

Answer

(0, 1) and (0, -1)

Question 22

A company produces two products, A and B. The profit per unit of product A is $5 and the profit per unit of product B is $8. The company has a limited production capacity represented by the constraint *2A + 3B ≤ 12*.  What is the maximum profit the company can make?

Accepted Answer

$32

Answer

$40

Answer

$24

Answer

$48

Question 23

Find the minimum value of the function *f(x,y) = x^2 + 2y^2* subject to the constraint *x + y = 1*.

Accepted Answer

2/3

Answer

4/3

Answer

1/3

Answer

1

Question 24

A farmer wishes to enclose a rectangular field using 100 meters of fencing. What is the maximum area of the field the farmer can enclose?

Accepted Answer

625 square meters

Answer

1000 square meters

Answer

2500 square meters

Answer

500 square meters

Question 25

Determine the point on the plane *x + 2y + 3z = 6* that is closest to the origin.

Accepted Answer

(1, 1, 1)

Answer

(1, 0, 2)

Answer

(2, 1, 0)

Answer

(0, 2, 1)

Question 26

A cylindrical can is to be constructed to hold 1 liter (1000 cubic centimeters) of liquid. Find the dimensions of the can that will minimize the amount of material used (i.e., minimize the surface area).

Accepted Answer

Radius = (1/√π) meters, Height = (2/√π) meters

Answer

Radius = (√π) meters, Height = (2√π) meters

Answer

Radius = (1/π) meters, Height = (2/π) meters

Answer

Radius = (2/√π) meters, Height = (1/√π) meters

Question 27

Find the maximum value of the function f(x,y) = xy subject to the constraint x^2 + y^2 = 1.

Accepted Answer

1/2

Answer

1/√2

Answer

1

Answer

√2/2

Question 28

A company manufactures two types of products, A and B. The production of each product requires the use of two resources, labor and material. The following table summarizes the resource requirements per unit of each product:

Accepted Answer

Produce 4 units of product A and 2 units of product B.

Answer

Produce 3 units of product A and 3 units of product B.

Answer

Produce 5 units of product A and 1 unit of product B.

Question 29

Find the critical points of the function f(x,y) = x^3 + y^3 - 3xy.

Accepted Answer

(0,0) and (1, 1)

Answer

(0,0) and (1, 0)

Answer

(1,0) and (0, 1)

Answer

(0,0) and (0, 1)