Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Lesson: Consequences of Differentiability

Worksheet • 12 Questions

Q1:

The graph of the first derivative 𝑓 β€² of a continuous function 𝑓 is shown. On what intervals is 𝑓 increasing or decreasing?

  • A 𝑓 is increasing on the intervals ( 0 , 2 ) , ( 4 , 6 ) , and ( 8 , ∞ ) and decreasing on the intervals ( 2 , 4 ) and ( 6 , 8 ) .
  • B 𝑓 is increasing on the intervals ( 1 , 5 ) and ( 6 , ∞ ) and decreasing on the intervals ( 0 , 1 ) and ( 5 , 6 ) .
  • C 𝑓 is increasing on the intervals ( 2 , 4 ) and ( 6 , 8 ) and decreasing on the intervals ( 0 , 2 ) , ( 4 , 6 ) , and ( 8 , ∞ ) .
  • D 𝑓 is increasing on the interval ( 0 , 3 ) and decreasing on the intervals ( 3 , 6 ) and ( 6 , ∞ ) .
  • E 𝑓 is increasing on the intervals ( 3 , 6 ) and ( 6 , ∞ ) and decreasing on the interval ( 0 , 3 ) .

Q2:

Determine the intervals on which the function is increasing and where it is decreasing.

  • Aincreasing over , decreasing over
  • Bdecreasing over , increasing over
  • C decreasing over
  • D increasing over

Q3:

Which of the following statements is true for the function ?

  • A is increasing on and decreasing on .
  • B is increasing on and decreasing on .
  • C is increasing on and decreasing on .
  • D is increasing on and decreasing on .
  • E is increasing on and decreasing on .

Q4:

Which of the following statements is true for the function ?

  • A is increasing on and decreasing on .
  • B is increasing on and decreasing on .
  • C is increasing on and decreasing on .
  • D is increasing on and decreasing on .
  • E is increasing on and decreasing on .

Q5:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 1 π‘₯ + 2 3 is concave up and down.

  • A The function is concave down on ( βˆ’ ∞ , 0 ) and concave up on ( 0 , ∞ ) .
  • B The function is concave down on ο€Ώ βˆ’ ∞ , βˆ’ √ 3 3 3  and concave up on ο€Ώ √ 3 3 3 , ∞  .
  • C The function is concave down on ( 0 , ∞ ) and concave up on ( βˆ’ ∞ , 0 ) .
  • D The function is concave down on ο€Ώ βˆ’ ∞ , βˆ’ √ 3 3 3  and ο€Ώ √ 3 3 3 , ∞  and concave up on ο€Ώ βˆ’ √ 3 3 3 , √ 3 3 3  .
  • E The function is concave down on ο€Ώ βˆ’ ∞ , √ 3 3 3  and concave up on ο€Ώ βˆ’ √ 3 3 3 , ∞  .

Q6:

The graph of a function 𝑦 = 𝑓 ( π‘₯ ) is shown. At which point are d d 𝑦 π‘₯ and d d 2 2 𝑦 π‘₯ both positive?

  • Apoint 𝐡
  • Bpoint 𝐸
  • Cpoint 𝐴
  • Dpoint 𝐷
  • Epoint 𝐢

Q7:

The graph of a function 𝑦 = 𝑓 ( π‘₯ ) is shown. At which point are d d 𝑦 π‘₯ and d d 2 2 𝑦 π‘₯ both negative?

  • Apoint 𝐸
  • Bpoint 𝐡
  • Cpoint 𝐴
  • Dpoint 𝐷
  • Epoint 𝐢

Q8:

Determine the intervals on which 𝑓 ( π‘₯ ) = βˆ’ π‘₯ + π‘₯ + 1 6 π‘₯ 3 2 is concave up and concave down.

  • A The function is concave up on the interval ο€Ό βˆ’ ∞ , 1 3  and concave down on the interval ο€Ό 1 3 , ∞  .
  • B The function is concave up on the interval ο€Ό βˆ’ ∞ , βˆ’ 1 3  and concave down on the interval ο€Ό βˆ’ 1 3 , ∞  .
  • C The function is concave up on the interval ο€Ό 1 3 , ∞  and concave down on the interval ο€Ό βˆ’ ∞ , 1 3  .
  • D The function is concave up on the interval ( βˆ’ 2 , ∞ ) and concave down on the interval ο€Ό βˆ’ ∞ , 8 3  .
  • E The function is concave up on the interval ο€Ό βˆ’ ∞ , 8 3  and concave down on the interval ( βˆ’ 2 , ∞ ) .

Q9:

Let 𝑓 ( π‘₯ ) = βˆ’ 4 π‘₯ + 5 π‘₯ + 1 2 π‘₯ 3 2 . Determine the intervals where this function is increasing and where it is decreasing.

  • A The function is decreasing on ο€Ό βˆ’ ∞ , βˆ’ 2 3  and ο€Ό 3 2 , ∞  and increasing on ο€Ό βˆ’ 2 3 , 3 2  .
  • B The function is decreasing on ο€Ό βˆ’ 3 2 , 2 3  and increasing on ο€Ό βˆ’ ∞ , βˆ’ 3 2  and ο€Ό 2 3 , ∞  .
  • C The function is decreasing on ο€Ό βˆ’ 2 3 , 3 2  and increasing on ο€Ό βˆ’ ∞ , βˆ’ 2 3  and ο€Ό 3 2 , ∞  .
  • D The function is decreasing on ο€Ό βˆ’ ∞ , βˆ’ 3 2  and ο€Ό 2 3 , ∞  and increasing on ο€Ό βˆ’ 3 2 , 2 3  .
  • E The function is decreasing on ο€Ό βˆ’ ∞ , βˆ’ 3 2  and ο€Ό 2 3 , ∞  and increasing on ο€Ό βˆ’ 3 2 , 2 3  .

Q10:

Let 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ + 3 π‘₯ + 1 2 π‘₯ 3 2 . Determine the intervals where this function is increasing and where it is decreasing.

  • A The function is decreasing on ( βˆ’ ∞ , βˆ’ 1 ) and ( 2 , ∞ ) and increasing on ( βˆ’ 1 , 2 ) .
  • B The function is decreasing on ( βˆ’ 2 , 1 ) and increasing on ( βˆ’ ∞ , βˆ’ 2 ) and ( 1 , ∞ ) .
  • C The function is decreasing on ( βˆ’ 1 , 2 ) and increasing on ( βˆ’ ∞ , βˆ’ 1 ) and ( 2 , ∞ ) .
  • D The function is decreasing on ( βˆ’ ∞ , βˆ’ 2 ) and ( 1 , ∞ ) and increasing on ( βˆ’ 2 , 1 ) .
  • E The function is decreasing on ( βˆ’ ∞ , βˆ’ 1 ) and ο€Ό 1 2 , ∞  and increasing on ο€Ό βˆ’ 1 , 1 2  .

Q11:

The graph of the first derivative 𝑓 β€² of a continuous function 𝑓 is shown. On what intervals is 𝑓 increasing or decreasing?

  • A 𝑓 is increasing on the intervals ( 1 , 6 ) and ( 8 , ∞ ) and decreasing on the intervals ( 0 , 1 ) and ( 6 , 8 ) .
  • B 𝑓 is increasing on the intervals ( 2 , 4 ) and ( 6 , ∞ ) and decreasing on the intervals ( 0 , 2 ) and ( 4 , 6 ) .
  • C 𝑓 is increasing on the intervals ( 0 , 1 ) and ( 6 , 8 ) and decreasing on the intervals ( 1 , 6 ) and ( 8 , ∞ ) .
  • D 𝑓 is increasing on the intervals ( 2 , 3 ) and ( 5 , 7 ) and decreasing on the intervals ( 0 , 2 ) , ( 3 , 5 ) , and ( 7 , ∞ ) .
  • E 𝑓 is increasing on the intervals ( 0 , 2 ) , ( 3 , 5 ) , and ( 7 , ∞ ) and decreasing on the intervals ( 2 , 3 ) and ( 5 , 7 ) .

Q12:

Which of the following statements is true for the function ?

  • AThe function is increasing on and decreasing on .
  • BThe function is decreasing on and increasing on .
  • CThe function is increasing on .
  • DThe function is decreasing on .
Preview