Worksheet: Increasing and Decreasing Intervals of a Function Using Derivatives

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In this worksheet, we will practice determining the increasing and decreasing intervals of functions using the first derivative of a function.

Q1:

Find all possible intervals on which the function is increasing and decreasing.

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

Q2:

Given that 𝑓 ( 𝑥 ) = 2 𝑥 + 2 𝑥 s i n c o s , 0 ≤ 𝑥 ≤ 𝜋 , determine the intervals on which 𝑓 is increasing or decreasing.

  • A 𝑓 is increasing on the intervals 3 𝜋 8 , 7 𝜋 8 and 0 , 3 𝜋 8 and decreasing on the interval 7 𝜋 8 , 𝜋 .
  • B 𝑓 is increasing on the interval 𝜋 8 , 5 𝜋 8 and decreasing on the intervals 0 , 𝜋 8 and 5 𝜋 8 , 𝜋 .
  • C 𝑓 is increasing on the interval 7 𝜋 8 , 𝜋 and decreasing on the intervals 3 𝜋 8 , 7 𝜋 8 and 0 , 3 𝜋 8 .
  • D 𝑓 is increasing on the intervals 0 , 𝜋 8 and 5 𝜋 8 , 𝜋 and decreasing on the interval 𝜋 8 , 5 𝜋 8 .
  • E 𝑓 is decreasing on the interval 0 , 5 𝜋 8 and decreasing on the interval 5 𝜋 8 , 𝜋 .

Q3:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = ( − 3 𝑥 − 1 2 ) 2 is increasing and on which it is decreasing.

  • Aincreasing on the interval ℝ
  • Bincreasing on the interval ] − ∞ , − 4 [ , decreasing on the interval ] − 4 , ∞ [
  • Cdecreasing on the interval ℝ
  • Ddecreasing on the interval ] − ∞ , − 4 [ , increasing on the interval ] − 4 , ∞ [

Q4:

The concentration 𝐶 of a drug in a patient’s bloodstream 𝑡 hours after injection is given by 𝐶 ( 𝑡 ) = 2 𝑡 3 + 𝑡 2 . How does the concentration 𝐶 change as 𝑡 increases?

  • AThe concentration 𝐶 of the drug does not change.
  • BThe concentration 𝐶 of the drug increases.
  • CThe concentration 𝐶 of the drug increases up to a certain point and then decreases.

Q5:

Determine the intervals on which is increasing or decreasing.

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

Q6:

Let 𝑓 ( 𝑥 ) = 3 𝑥 𝑒 4 − 4 𝑥 . Determine the intervals where this function is increasing and where it is decreasing.

  • A 𝑓 is increasing on the intervals − ∞ , 1 2 and 3 2 , ∞ and decreasing on the interval 1 2 , 3 2 .
  • B 𝑓 is increasing on the intervals ] − ∞ , 0 [ and ] 1 , ∞ [ and decreasing on the interval ] 0 , 1 [ .
  • C 𝑓 is increasing on the interval 3 2 , ∞ and decreasing on the intervals 1 2 , 3 2 and − ∞ , 1 2 .
  • D 𝑓 is increasing on the interval ] 0 , 1 [ and decreasing on the intervals ] − ∞ , 0 [ and ] 1 , ∞ [ .
  • E 𝑓 is decreasing on the interval ] − 1 , 0 [ and decreasing on the intervals ] − ∞ , − 1 [ and ] 0 , ∞ [ .

Q7:

Given that , find the intervals on which is increasing or decreasing.

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

Q8:

Find the intervals on which the function is increasing and decreasing.

  • A The function is decreasing on .
  • BThe function is increasing on .
  • CThe function is increasing on .
  • D The function is decreasing on .
  • E The function is decreasing on .

Q9:

Find the intervals on which the function is increasing and decreasing.

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

Q10:

Find the intervals on which the function is increasing and decreasing.

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

Q11:

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

  • AThe function is decreasing on and and increasing on .
  • BThe function is decreasing on and increasing on and .
  • CThe function is decreasing on and increasing on and .
  • D The function is decreasing on and and increasing on .
  • EThe function is decreasing on and increasing on and .

Q12:

Find the intervals on which the function 𝑓 ( 𝑥 ) = 2 𝑒 − 3 𝑒 + 4 𝑥 𝑥 is either increasing or decreasing.

  • AThe function is increasing on ] − ∞ , ∞ [ .
  • BThe function is decreasing on − ∞ , 4 3 l n and 4 3 , ∞ l n .
  • CThe function is decreasing on ] − ∞ , ∞ [ .
  • DThe function is increasing on − ∞ , 4 3 l n and 4 3 , ∞ l n .
  • EThe function is increasing on − ∞ , 3 4 l n and 3 4 , ∞ l n .

Q13:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = ( 𝑥 + 3 ) | 𝑥 + 3 | is increasing and decreasing.

  • Aincreasing over ℝ − { − 3 }
  • Bincreasing over ℝ − { 3 }
  • Cdecreasing over ℝ − { − 3 }
  • Dincreasing over ℝ
  • Eincreasing over ] − ∞ , − 3 [ , decreasing over ] − 3 , ∞ [

Q14:

Determine the intervals over which the function 𝑓 ( 𝑥 ) = − | 2 𝑥 | + 2 8 is increasing and over which it is decreasing.

  • Aincreasing over ℝ − { 1 4 }
  • Bdecreasing over the interval ] − ∞ , 0 [ , increasing over the interval ] 0 , ∞ [
  • Cdecreasing over ℝ − { 1 4 }
  • D decreasing over the interval ] 0 , ∞ [ , increasing over the interval ] − ∞ , 0 [
  • Edecreasing over the interval ] 1 4 , ∞ [ , increasing over the interval ] − ∞ , 1 4 [

Q15:

Let Find the intervals on which 𝑓 is increasing and where it is decreasing.

  • Aincreasing over the interval ] − ∞ , 1 [ , decreasing over the interval ] 1 , ∞ [
  • Bincreasing over the interval ] − ∞ , 0 [ , decreasing over the interval ] 0 , ∞ [
  • Cincreasing over ℝ − { 1 }
  • Ddecreasing over the interval ] − ∞ , 0 [ , increasing over the interval ] 0 , ∞ [
  • Edecreasing over ℝ − { 1 }

Q16:

Determine the intervals on which the function 𝑦 = 3 𝑥 ( 9 𝑥 + 5 ) 2 is increasing and where it is decreasing.

  • A increasing over ℝ
  • B increasing over the interval − 1 0 2 7 , 0 , decreasing over the intervals − ∞ , − 1 0 2 7 and ] 0 , ∞ [
  • C decreasing over ℝ
  • D increasing over the intervals − ∞ , − 1 0 2 7 and ] 0 , ∞ [ , decreasing over the interval − 1 0 2 7 , 0

Q17:

Determine the intervals over which the function 𝑓 ( 𝑥 ) = 1 1 𝑥 − 8 𝑥 3 2 is increasing and over which it is decreasing.

  • Aincreasing over ℝ − { 0 }
  • Bincreasing over the interval ℝ − 1 6 3 3
  • Cincreasing over ℝ − 1 6 3 3
  • Ddecreasing over the interval 0 , 1 6 3 3 , increasing over the intervals ] − ∞ , 0 [ and 1 6 3 3 , ∞
  • Eincreasing over the interval 0 , 1 6 3 3 , decreasing over the intervals ] − ∞ , 0 [ and 1 6 3 3 , ∞

Q18:

Given that 𝑓 ( 𝑥 ) = 8 𝑥 − 1 6 𝑥 + 5 4 2 , determine the intervals on which 𝑓 is increasing or decreasing.

  • A 𝑓 is increasing on the intervals ] − 1 , 0 [ and ] − ∞ , − 1 [ and decreasing on the intervals ] 1 , ∞ [ and ] 0 , 1 [ .
  • B 𝑓 is increasing on the intervals ] − ∞ , − 1 [ and ] 0 , 1 [ and decreasing on the intervals ] − 1 , 0 [ and ] 1 , ∞ [ .
  • C 𝑓 is increasing on the intervals ] 1 , ∞ [ and ] − ∞ , − 1 [ and decreasing on the intervals ] − 1 , 0 [ and ] 1 , ∞ [ .
  • D 𝑓 is increasing on the intervals ] − 1 , 0 [ and ] 1 , ∞ [ and decreasing on the intervals ] − ∞ , − 1 [ and ] 0 , 1 [ .
  • E 𝑓 is increasing on the intervals ] − 1 , 0 [ and ] 0 , 1 [ and decreasing on the intervals ] − ∞ , − 1 [ and ] 1 , ∞ [ .

Q19:

Determine the intervals on which the function is increasing or decreasing.

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

Q20:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 3 𝑥 − 9 𝑥 − 4 3 2 is increasing and on which it is decreasing.

  • A 𝑓 is increasing on ] − ∞ , 0 [ and decreasing on ] 2 , ∞ [ .
  • B 𝑓 is decreasing on ] − ∞ , 0 [ , ] 2 , ∞ [ and increasing on ] 0 , 2 [ .
  • C 𝑓 is decreasing on ] − ∞ , 0 [ and increasing on ] 2 , ∞ [ .
  • D 𝑓 is increasing on ] − ∞ , 0 [ , ] 2 , ∞ [ and decreasing on ] 0 , 2 [ .

Q21:

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

  • A The function is increasing on and decreasing on .
  • B The function is decreasing on .
  • CThe function is increasing on and decreasing on .
  • D The function is increasing on .
  • E The function is increasing on and decreasing on .

Q22:

For , find the intervals on which is increasing or decreasing.

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

Q23:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 7 𝑥 𝑥 + 9 2 is increasing and where it is decreasing.

  • Aincreasing on the interval ] − ∞ , − 3 [ , decreasing on the interval ] 3 , ∞ [
  • Bincreasing on the intervals ] − ∞ , − 3 [ and ] 3 , ∞ [ , decreasing on the interval ] − 3 , 3 [
  • Cdecreasing on the interval ] − ∞ , − 3 [ , increasing on the interval ] 3 , ∞ [
  • Ddecreasing on the intervals ] − ∞ , − 3 [ and ] 3 , ∞ [ , increasing on the interval ] − 3 , 3 [

Q24:

Determine the intervals on which the function 𝑓 ( 𝑥 ) = 8 𝑥 − 7 7 𝑥 − 5 is increasing and where it is decreasing.

  • Adecreasing on the interval − ∞ , 5 7 , increasing on the interval 5 7 , ∞
  • Bdecreasing over ℝ − 5 7
  • Cdecreasing over ℝ
  • Dincreasing over ℝ − 5 7

Q25:

Determine the intervals on which the function 𝑦 = 7 𝑥 𝑥 − 8 is increasing and on which it is decreasing.

  • A decreasing on ] − ∞ , 8 [ , increasing on ] 8 , ∞ [
  • B decreasing on ] 8 , ∞ [ , increasing on ] − ∞ , 8 [
  • C decreasing on ℝ
  • D decreasing on ℝ − { 8 }
  • E increasing on ℝ

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