Optimization in Calculus I: Master Finding Max and Min Values

Question 1

Find the critical point(s) of the function f(x) = x³ - 3x² + 2.

Accepted Answer

x = 0, x = 2

Answer

x = 0

Answer

x = 2

Answer

x = 1, x = 2

Question 2

Determine whether the function f(x) = -x² + 4x - 3 has a maximum or minimum at x = 2.

Accepted Answer

Minimum

Answer

Maximum

Answer

Neither a maximum nor a minimum

Answer

Cannot be determined

Question 3

Which of the following functions has a global maximum at x = 0?

Accepted Answer

f(x) = -x² + 2x + 1

Answer

f(x) = x² - 4

Answer

f(x) = |x|

Answer

f(x) = sin(x)

Question 4

Which of the following optimization problems can be solved using the method of bisection?

Accepted Answer

Solving an equation f(x) = 0

Answer

Maximizing a function over a closed interval

Answer

Minimizing a function over an unbounded domain

Answer

Finding the intersection point of two curves

Question 5

Determine the intervals where the function f(x) = x^4 - 2x^2 + 1 is concave upward and concave downward. Also, identify the x-coordinates of any inflection points.

Accepted Answer

Concave upward on (-∞, -1) and (1, ∞), concave downward on (-1, 1); inflection points at x = -1 and x = 1

Answer

Concave upward on (-∞, -1) and (0, ∞), concave downward on (-1, 0); inflection points at x = -1 and x = 0

Answer

Concave upward on (-1, 1), concave downward on (-∞, -1) and (1, ∞); inflection points at x = -1 and x = 1

Answer

Concave upward on (-∞, -2) and (2, ∞), concave downward on (-2, 2); inflection points at x = -2 and x = 2

Question 6

To find the critical points of a function f(x), you need to set its first derivative equal to:

Accepted Answer

0

Answer

1

Question 7

The second derivative test is used to determine the:

Accepted Answer

Concavity of a function

Answer

Finding the critical points of a function

Question 8

If the second derivative of a function f(x) is positive at a critical point c, then the function has a:

Accepted Answer

Local maximum at c

Answer

Local minimum at c

Question 9

A function f(x) is concave up on an interval where its graph is:

Accepted Answer

Curving upwards

Answer

Curving downwards

Question 10

Finding the absolute maximum and minimum values of a function on a closed interval is known as:

Accepted Answer

Global optimization

Answer

Local optimization

Question 11

A function f(x) has a critical point at x = 0 where f'(0) = 0 and f''(0) < 0. This indicates that the function has a:

Accepted Answer

Saddle point at x = 0

Answer

Local minimum

Question 12

A rectangular garden with a perimeter of 100 feet has the maximum area of:

Accepted Answer

625 square feet

Answer

1250 square feet

Question 13

Which of the following is the second derivative test?

Accepted Answer

If f''(x) > 0, then x is a local minimum. If f''(x) < 0, then x is a local maximum.

Answer

If f'(x) = 0 and f''(x) = 0, then x is neither a local minimum nor a local maximum.

Answer

If f'''(x) = 0, then x is an inflection point.

Question 14

A farmer has 100 feet of fencing to enclose a rectangular garden. What dimensions will result in the largest possible area?

Accepted Answer

50 feet by 25 feet

Answer

100 feet by 5 feet

Answer

25 feet by 25 feet

Answer

75 feet by 12.5 feet

Question 15

Which of the following is a necessary condition for a function to have a local maximum or minimum?

Accepted Answer

The first derivative is zero at the point.

Answer

The third derivative is zero at the point.

Answer

The second derivative is positive at the point.

Answer

The second derivative is negative at the point.

Question 16

A farmer has 100 meters of fencing to enclose a rectangular plot of land. The plot has two perpendicular fences parallel to two adjacent sides of the rectangular plot. Determine the dimensions of the rectangular plot that maximize the enclosed area.

Accepted Answer

50 meters by 25 meters

Answer

25 meters by 25 meters

Answer

100 meters by 1 meter

Question 17

Find the value of c that makes the function f(x) = x² + cx - 3 have a relative maximum at x = 1.

Accepted Answer

c = 2

Answer

c = 0

Answer

c = -2

Answer

c = -4

Question 18

Find the critical point(s) of the function f(x) = |x - 1|.

Accepted Answer

x = 1

Answer

x = 2

Answer

x = 0

Answer

f(x) has no critical points

Question 19

Find the minimum value of f(x, y) = x² + y² on the circle x² + y² = 25.

Accepted Answer

0

Answer

5

Answer

-25

Answer

1

Question 20

What is the main objective of optimization?

Accepted Answer

Finding the maximum or minimum values of a function

Answer

Determining the area under a curve

Answer

Calculating the slope of a line

Question 21

Which mathematical tool is used to identify critical points of a function?

Accepted Answer

Derivatives

Answer

Limits

Answer

Second derivatives

Answer

Integrals

Question 22

Identify an example of an optimization problem in the real world:

Accepted Answer

Minimizing fuel consumption during a road trip

Answer

Finding the area of a triangle

Answer

Calculating the volume of a sphere

Answer

Solving a system of linear equations

Question 23

What is the definition of the absolute maximum of a function on a closed interval?

Accepted Answer

The largest value the function assumes within the interval

Answer

The average value of the function over the interval

Answer

The smallest value the function takes on the interval

Question 24

Describe the process for finding the global maximum of a function on an open interval:

Accepted Answer

Locate all critical points and evaluate the function at the interval's endpoints

Answer

Employ the second derivative test

Answer

Apply the mean value theorem

Question 25

Which condition must be satisfied for a point to be considered a critical point?

Accepted Answer

The first derivative of the function at that point is zero

Answer

The third derivative of the function at that point is zero

Answer

The second derivative of the function at that point is zero

Answer

The function must be differentiable at that point

Question 26

What is the point of diminishing returns?

Accepted Answer

The point at which the rate of change of a function starts decreasing

Answer

The point at which the rate of change of a function is minimum

Answer

The point at which the rate of change of a function is maximum

Question 27

Identify a primary benefit of utilizing optimization techniques:

Accepted Answer

Enhancing efficiency and maximizing performance

Answer

Reducing risk to zero

Answer

Guaranteeing absolute success

Answer

Eliminating all uncertainties

Question 28

Find the second derivative of the function f(x) = sin(x) + cos(x).

Accepted Answer

-sin(x) - cos(x)

Answer

sin(x) + cos(x)

Answer

sin(x) - cos(x)

Answer

-sin(x) + cos(x)

Question 29

What point on the curve y = x² - 4x + 3 is closest to the point (3, 0)?

Accepted Answer

(2, -1)

Answer

(4, 5)

Answer

(1, 0)

Answer

(3, 0)

Question 30

What is the maximum value of the function *f(x) = e^(-x^2)* on the interval [-1, 1]?

Accepted Answer

e^(-1)

Answer

e

Answer

The maximum value does not exist.

Answer

1

Question 31

A farmer wants to build a rectangular fence enclosing 100 square meters of land along a straight river. The river will act as one side of the enclosure.  What is the minimum length of fencing the farmer needs?

Accepted Answer

40 meters

Answer

50 meters

Answer

20 meters

Answer

100 meters

Question 32

Which condition is NOT required for a function f(x) to have a local maximum at x = a?

Accepted Answer

f''(a) > 0

Answer

f(x) is defined on an open interval containing a

Answer

f'(x) changes sign from positive to negative at x = a

Answer

f'(a) = 0

Question 33

What is the absolute maximum value of the function f(x) = x³ - 3x + 2 on the closed interval [0, 2]?

Accepted Answer

4

Answer

0

Answer

2

Answer

6

Question 34

What are the critical points of the function *g(x) = (x - 2)²(x + 1)*?

Accepted Answer

x = 2 and x = -1

Answer

x = 2 only

Answer

There are no critical points.

Answer

x = -1 only

Question 35

If a function *h(x)* has a local minimum at *x = d*, what is true about the function's behavior at that point?

Accepted Answer

The function changes from decreasing to increasing at *x = d*.

Answer

*h(d)* is the absolute minimum value of the function.

Answer

*h''(d)* must be negative.

Answer

*h'(d)* must be positive.

Question 36

What is the maximum area of a rectangle that can be inscribed in a semicircle with a radius of 5 units?

Accepted Answer

25 square units

Answer

50 square units

Answer

50π square units

Answer

12.5 square units

Question 37

A particle moves along a straight line with its position function given by *s(t) = t³ - 6t² + 9t + 2*. At what times is the particle at rest?

Accepted Answer

t = 1 and t = 3

Answer

t = 1 only

Answer

t = 3 only

Answer

t = 0 and t = 2

Question 38

A rectangular garden is to be enclosed by a fence on three sides, with the house forming the fourth side. If 50 meters of fencing is available, what are the dimensions of the garden that maximize its area?

Accepted Answer

25 meters by 12.5 meters

Answer

10 meters by 20 meters

Answer

20 meters by 10 meters

Answer

15 meters by 15 meters

Question 39

What are the critical points of the function *f(x) = x^3 - 3x^2 + 2*?

Accepted Answer

x = 0 and x = 2

Answer

x = -1 and x = 1

Answer

x = -2 and x = 0

Answer

x = 1 and x = 2

Question 40

What is the minimum value of the function *f(x) = x^2 - 4x + 5* on the interval [0, 3]?

Accepted Answer

1

Answer

2

Answer

0

Answer

3

Question 41

A company produces and sells *x* units of a product. The cost function is *C(x) = 100 + 2x*, and the revenue function is *R(x) = 10x - 0.1x^2*. What production level maximizes the company's profit?

Accepted Answer

x = 40

Answer

x = 30

Answer

x = 20

Answer

x = 50

Question 42

Which of the following is a necessary condition for a function *f(x)* to have a local maximum at *x = c*?

Accepted Answer

*f'(c) = 0* or *f'(c)* does not exist

Answer

*f''(c) < 0*

Answer

*f(c) > f(x)* for all *x* near *c*

Answer

*f''(c) > 0*

Question 43

Determine the nature of the critical point x = 1 for the function f(x) = x^3 - 3x + 2.

Accepted Answer

Local maximum

Answer

Local minimum

Answer

Saddle point

Answer

None of the above

Question 44

A cylindrical can is to hold 1 liter of liquid. Find the dimensions of the can that minimize the amount of material used.

Accepted Answer

Radius = 5.42 cm, Height = 10.84 cm

Answer

Radius = 4 cm, Height = 8 cm

Answer

Radius = 6 cm, Height = 12 cm

Answer

Radius = 7 cm, Height = 14 cm

Question 45

Find the absolute maximum and minimum values of the function f(x) = x^4 - 2x^2 + 1 on the interval [-2, 2].

Accepted Answer

Absolute maximum: 9, Absolute minimum: 0

Answer

Absolute maximum: 1, Absolute minimum: 0

Answer

Absolute maximum: 9, Absolute minimum: -1

Answer

Absolute maximum: 1, Absolute minimum: -1

Question 46

What is the derivative of the function f(x) = sin(x^2) with respect to x?

Accepted Answer

2x cos(x^2)

Answer

cos(2x)

Answer

cos(x^2)

Answer

2x sin(x^2)

Question 47

The graph of a function f(x) has a horizontal tangent line at x = 2. What can we conclude about f'(2)?

Accepted Answer

f'(2) = 0

Answer

f'(2) does not exist

Answer

f'(2) < 0

Answer

f'(2) > 0