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06 Applications of Derivatives Quiz Online Quiz Questions

Use this free practice quiz with 20 questions to review 06 Applications of Derivatives, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

Which statement correctly defines a critical number of a function?

  1. A

    A critical number is any value where the function itself is undefined.

  2. B

    A critical number is a domain value where the derivative is zero or does not exist.

  3. C

    Every critical number must be a local maximum or local minimum.

  4. D

    A critical number can occur only when the second derivative is zero.

02
True or false
1 point

True or false: To apply the Mean Value Theorem on [a,b], a function must be continuous on [a,b] and differentiable on (a,b).

  1. A

    True

  2. B

    False

03
Fill in the blank
1 point

Complete the First Derivative Test statements: If f' changes from positive to negative at a critical number, the function has a there. If f' changes from negative to positive, it has a there.

04
Choose one
1 point

For f(x)=x^2 on the interval [1,3], which value of c satisfies the Mean Value Theorem equation f'(c) = [f(3)-f(1)]/(3-1)?

  1. A

    c=1

  2. B

    c=3

  3. C

    c=2

  4. D

    No such c exists

05
Written response
1 point

For f(x)=x^3-3x^2-9x+5, enter the x-value where f''(x)=0 and the concavity changes.

06
Choose all
1 point

Select all conclusions that are supported by the derivative information given.

  1. A

    If f'(x)>0 throughout an interval, f is increasing there.

  2. B

    If f'(x)<0 throughout an interval, f is decreasing there.

  3. C

    If f'(x)=0 throughout an interval, f is constant there.

  4. D

    If f'(c)=0 at one point, f must be constant on the entire domain.

07
True or false
1 point

True or false: Because f'(0)=0 for f(x)=x^3, the function has a local maximum or local minimum at x=0.

  1. A

    True

  2. B

    False

08
Fill in the blank
1 point

Complete the concavity rules: If f''(x)>0 on an interval, the graph is there. If f''(x)<0, it is there.

09
Choose one
1 point

Suppose f'(c)=0, f'' is continuous near c, and f''(c)>0. What conclusion follows from the Second Derivative Test?

  1. A

    f has a local minimum at c

  2. B

    f has a local maximum at c

  3. C

    f has neither a local maximum nor a local minimum at c

  4. D

    The second derivative test gives no information in this case

10
Written response
1 point

What is the mathematical term for a point on a graph where the concavity changes?

11
Choose all
1 point

Select all steps that belong to the recommended procedure for sketching a function using algebraic and derivative information.

  1. A

    Determine the domain.

  2. B

    Find the intercepts.

  3. C

    Check for even or odd symmetry.

  4. D

    Examine end behavior and asymptotes.

  5. E

    Assume every point where f''(x)=0 is an inflection point without checking a sign change.

12
Open ended
1 point

For f(x)=x^3-3x, determine the critical numbers and classify each as a local maximum or local minimum. Give the coordinates of both extrema, then determine the concavity intervals and the inflection point.

13
Choose one
1 point

When finding absolute extrema of a continuous function on a closed interval [a,b], what additional step is essential after finding interior critical numbers?

  1. A

    Endpoints can never be absolute extrema because they are not interior critical numbers.

  2. B

    The endpoint values must be checked along with values at interior critical numbers.

  3. C

    Only points where f'(x) is undefined need to be checked.

  4. D

    The second derivative alone always identifies the absolute maximum and minimum.

14
True or false
1 point

True or false: If f''(c)=0, then (c,f(c)) is automatically an inflection point.

  1. A

    True

  2. B

    False

15
Choose one
1 point

Which value is a critical number of f(x) = |x - 2|?

  1. A

    x = 0

  2. B

    x = 1

  3. C

    x = 2

  4. D

    x = 3

16
Written response
1 point

For f(x) = x^2 on [1, 5], use the Mean Value Theorem to find the value of c in (1, 5) such that f'(c) equals the average rate of change on the interval.

17
Written response
1 point

For f(x) = x^3 - 6x^2 + 9x, enter the x-value where the second derivative is zero and a change in concavity may occur.

18
Choose one
1 point

At a critical number c, suppose f is continuous near c and f' changes from negative immediately to the left of c to positive immediately to the right. What can be concluded?

  1. A

    A local maximum

  2. B

    A local minimum

  3. C

    An inflection point

  4. D

    No conclusion is possible

19
Choose one
1 point

Suppose f'(c) = 0 and f''(c) > 0, with f'' continuous near c. What conclusion follows from the Second Derivative Test?

  1. A

    f has a local minimum at c

  2. B

    f has a local maximum at c

  3. C

    f has an inflection point at c

  4. D

    f is constant near c

20
Choose one
1 point

Which point is the inflection point of f(x) = x^3 - 3x?

  1. A

    (-1, 2)

  2. B

    (0, -3)

  3. C

    (0, 0)

  4. D

    (3, 0)