Worksheet: Graphing Using Derivative

In this worksheet, we will practice drawing the curve of a function by finding its critical and inflection points and finding the intervals where the function is decreasing and increasing.

Q1:

Which of the following is the graph of 𝑓 ( π‘₯ ) = βˆ’ ( π‘₯ + 1 ) ( π‘₯ βˆ’ 2 ) 3 ?

  • A
  • B
  • C
  • D
  • E

Q2:

Use the given graph of a function 𝑓 β€² β€² to find the π‘₯ -coordinates of the inflection points of 𝑓 .

  • A 𝑓 has inflection points at π‘₯ = 3 and π‘₯ = 5 .
  • B 𝑓 has inflection points at π‘₯ = 2 , π‘₯ = 4 , and π‘₯ = 6 .
  • C 𝑓 has inflection points at π‘₯ = 2 and π‘₯ = 6 .
  • D 𝑓 has inflection points at π‘₯ = 1 and π‘₯ = 7 .
  • E 𝑓 has inflection points at π‘₯ = 4 and π‘₯ = 6 .

Q3:

The graph of the first derivative 𝑓 β€² of a continuous function 𝑓 is shown. State the π‘₯ -coordinates of the inflection points of 𝑓 .

  • A 𝑓 has inflection points at π‘₯ = 1 , π‘₯ = 6 , and π‘₯ = 8 .
  • B 𝑓 has inflection points at π‘₯ = 2 and π‘₯ = 6 .
  • C 𝑓 has inflection points at π‘₯ = 0 , π‘₯ = 1 , π‘₯ = 6 , and π‘₯ = 8 .
  • D 𝑓 has inflection points at π‘₯ = 2 , π‘₯ = 3 , π‘₯ = 5 , and π‘₯ = 7 .
  • E 𝑓 has inflection points at π‘₯ = 2 . 5 and π‘₯ = 4 .

Q4:

The graph of the first derivative 𝑓 β€² of a function 𝑓 is shown. What are the π‘₯ -coordinates of the inflection points of 𝑓 ?

  • A 𝑓 has inflection points at π‘₯ = 0 and π‘₯ = 9 .
  • B 𝑓 has inflection points at π‘₯ = 4 , π‘₯ = 6 , and π‘₯ = 8 .
  • C 𝑓 has inflection points at π‘₯ = 4 and π‘₯ = 6 .
  • D 𝑓 has inflection points at π‘₯ = 1 , π‘₯ = 2 , π‘₯ = 3 , π‘₯ = 5 , and π‘₯ = 7 .
  • E 𝑓 has inflection points at π‘₯ = 1 . 5 , π‘₯ = 2 . 5 , π‘₯ = 4 , and π‘₯ = 6 .

Q5:

Using the graph, discuss the monotony of the function.

  • A 𝑓 ( π‘₯ ) is increasing on the interval ( βˆ’ ∞ , 0 ) and decreasing on the interval ( 0 , ∞ ) .
  • B 𝑓 ( π‘₯ ) is increasing on the interval ( 0 , ∞ ) and decreasing on the interval ( βˆ’ ∞ , 0 ) .
  • C 𝑓 ( π‘₯ ) is increasing on the interval ( 2 , ∞ ) and decreasing on the interval ( βˆ’ ∞ , 2 ) .
  • D 𝑓 ( π‘₯ ) is increasing on ℝ .

Q6:

Find where (if at all) the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 5 π‘₯ βˆ’ 1 5 π‘₯ + 1 2 has its local maxima and minima.

  • Alocal minimum at π‘₯ = 0 , no local maximum
  • Blocal minimum at π‘₯ = βˆ’ 1 3 , local maximum at π‘₯ = 2 8
  • Clocal maximum at π‘₯ = 1 , no local minimum
  • Dlocal maximum at π‘₯ = βˆ’ 2 , local minimum π‘₯ = 0

Q7:

Given that the curve π‘₯ 𝑦 + π‘Ž π‘₯ + 𝑏 𝑦 = 0 2 has an inflection point at ( 3 , βˆ’ 2 ) , what are the values of constants π‘Ž and 𝑏 ?

  • A π‘Ž = βˆ’ 4 , 𝑏 = 3
  • B π‘Ž = βˆ’ 8 , 𝑏 = 3
  • C π‘Ž = 2 4 , 𝑏 = 3
  • D π‘Ž = 8 , 𝑏 = 3

Q8:

Using the given graph of the function 𝑓 , at what values of π‘₯ does 𝑓 have inflection points?

  • A 𝑓 has inflection points when π‘₯ = 1 and π‘₯ = 7 .
  • B 𝑓 has inflection points when π‘₯ = 2 , π‘₯ = 4 , and π‘₯ = 6 .
  • C 𝑓 has inflection points when π‘₯ = 2 and π‘₯ = 6 .
  • D 𝑓 has inflection points when π‘₯ = 3 and π‘₯ = 5 .
  • E 𝑓 has inflection points when π‘₯ = 4 and π‘₯ = 6 .

Q9:

Using the given graph of the function 𝑓 β€² , at what values of π‘₯ does 𝑓 have inflection points?

  • A 𝑓 has inflection points when π‘₯ = 1 and π‘₯ = 7 .
  • B 𝑓 has inflection points when π‘₯ = 3 and π‘₯ = 5 .
  • C 𝑓 has inflection points when π‘₯ = 2 and π‘₯ = 6 .
  • D 𝑓 has inflection points when π‘₯ = 2 , π‘₯ = 4 and π‘₯ = 6 .
  • E 𝑓 has inflection points when π‘₯ = 4 and π‘₯ = 6 .

Q10:

The graph of the first derivative 𝑓 β€² of a continuous function 𝑓 is shown. State the π‘₯ -coordinates of the inflection points of 𝑓 .

  • A 𝑓 has inflection points at π‘₯ = 2 , π‘₯ = 4 , and π‘₯ = 8 .
  • B 𝑓 has inflection points at π‘₯ = 1 and π‘₯ = 5 .
  • C 𝑓 has an inflection point at π‘₯ = 6 .
  • D 𝑓 has an inflection point at π‘₯ = 3 .
  • E 𝑓 has inflection points at π‘₯ = 2 and π‘₯ = 4 .

Q11:

The graph of the derivative 𝑓 β€² of a function 𝑓 is shown. At what values of π‘₯ does 𝑓 have a local maximum or minimum?

  • A 𝑓 has a local maximum at π‘₯ = 3 .
  • B 𝑓 has a local maximum at π‘₯ = 1 and a local minimum at π‘₯ = 5 .
  • C 𝑓 has a local minimum at π‘₯ = 3 .
  • D 𝑓 has a local maximum at π‘₯ = 5 and a local minimum at π‘₯ = 1 .
  • E 𝑓 has a local maximum at π‘₯ = 0 and a local minimum at π‘₯ = 6 .

Q12:

The graph of the derivative of a function is shown. On what intervals is increasing or decreasing?

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

Q13:

Use the given graph of to find all possible intervals on which is increasing.

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

Q14:

The graph of the derivative of a function is shown. On what intervals is increasing or decreasing?

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

Q15:

Use the given graph of to find all possible intervals on which is decreasing.

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

Q16:

Use the given graph of to find all possible intervals on which is increasing.

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

Q17:

Using the graph, determine the intervals of increase and decrease of the function.

  • A 𝑓 ( π‘₯ ) is increasing on the interval ] βˆ’ 7 , ∞ [ and decreasing on the interval ] βˆ’ ∞ , βˆ’ 7 [ .
  • B 𝑓 ( π‘₯ ) is increasing on the interval ] βˆ’ 6 , ∞ [ and decreasing on the interval ] βˆ’ ∞ , βˆ’ 6 [ .
  • C 𝑓 ( π‘₯ ) is increasing on the interval ] βˆ’ ∞ , βˆ’ 7 [ and decreasing on the interval ] βˆ’ 7 , ∞ [ .
  • D 𝑓 ( π‘₯ ) is increasing on the interval ] βˆ’ ∞ , βˆ’ 6 [ and decreasing on the interval ] βˆ’ 6 , ∞ [ .
  • E 𝑓 ( π‘₯ ) is increasing on ℝ .

Q18:

Consider the polynomial function whose graph is given below.

Use the given points and the fact that is a critical point of the function to determine .

  • A
  • B
  • C
  • D
  • E

Determine the intervals where .

  • A
  • B
  • C
  • D
  • E

Q19:

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

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