Worksheet: Increasing and Decreasing Intervals of a Function Using Derivatives

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 ( π‘₯ ) = 2 π‘₯ + 2 π‘₯ s i n c o s , 0 ≀ π‘₯ ≀ πœ‹ , determine the intervals on which 𝑓 is increasing or decreasing.

  • A 𝑓 is increasing on the intervals  3 πœ‹ 8 , 7 πœ‹ 8  and  0 , 3 πœ‹ 8  and decreasing on the interval  7 πœ‹ 8 , πœ‹  .
  • B 𝑓 is increasing on the interval  πœ‹ 8 , 5 πœ‹ 8  and decreasing on the intervals  0 , πœ‹ 8  and  5 πœ‹ 8 , πœ‹  .
  • C 𝑓 is increasing on the interval  7 πœ‹ 8 , πœ‹  and decreasing on the intervals  3 πœ‹ 8 , 7 πœ‹ 8  and  0 , 3 πœ‹ 8  .
  • D 𝑓 is increasing on the intervals  0 , πœ‹ 8  and  5 πœ‹ 8 , πœ‹  and decreasing on the interval  πœ‹ 8 , 5 πœ‹ 8  .
  • E 𝑓 is decreasing on the interval  0 , 5 πœ‹ 8  and decreasing on the interval  5 πœ‹ 8 , πœ‹  .

Q3:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = ( βˆ’ 3 π‘₯ βˆ’ 1 2 ) 2 is increasing and on which it is decreasing.

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

Q4:

The concentration 𝐢 of a drug in a patient’s bloodstream 𝑑 hours after injection is given by 𝐢 ( 𝑑 ) = 2 𝑑 3 + 𝑑 2 . How does the concentration 𝐢 change as 𝑑 increases?

  • AThe concentration 𝐢 of the drug does not change.
  • BThe concentration 𝐢 of the drug increases.
  • CThe concentration 𝐢 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 ( π‘₯ ) = 3 π‘₯ 𝑒 4 βˆ’ 4 π‘₯ . Determine the intervals where this function is increasing and where it is decreasing.

  • A 𝑓 is increasing on the intervals  βˆ’ ∞ , 1 2  and  3 2 , ∞  and decreasing on the interval  1 2 , 3 2  .
  • B 𝑓 is increasing on the intervals ] βˆ’ ∞ , 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  βˆ’ ∞ , 1 2  .
  • D 𝑓 is increasing on the interval ] 0 , 1 [ and decreasing on the intervals ] βˆ’ ∞ , 0 [ and ] 1 , ∞ [ .
  • E 𝑓 is decreasing on the interval ] βˆ’ 1 , 0 [ and decreasing on the intervals ] βˆ’ ∞ , βˆ’ 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 ( π‘₯ ) = 2 𝑒 βˆ’ 3 𝑒 + 4 π‘₯ π‘₯ is either increasing or decreasing.

  • AThe function is increasing on ] βˆ’ ∞ , ∞ [ .
  • BThe function is decreasing on  βˆ’ ∞ , ο€Ό 4 3   l n and  ο€Ό 4 3  , ∞  l n .
  • CThe function is decreasing on ] βˆ’ ∞ , ∞ [ .
  • DThe function is increasing on  βˆ’ ∞ , ο€Ό 4 3   l n and  ο€Ό 4 3  , ∞  l n .
  • EThe function is increasing on  βˆ’ ∞ , ο€Ό 3 4   l n and  ο€Ό 3 4  , ∞  l n .

Q13:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = ( π‘₯ + 3 ) | π‘₯ + 3 | is increasing and decreasing.

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

Q14:

Determine the intervals over which the function 𝑓 ( π‘₯ ) = βˆ’ | 2 π‘₯ | + 2 8 is increasing and over which it is decreasing.

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

Q15:

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

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

Q16:

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

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

Q17:

Determine the intervals over which the function 𝑓 ( π‘₯ ) = 1 1 π‘₯ βˆ’ 8 π‘₯ 3 2 is increasing and over which it is decreasing.

  • Aincreasing over ℝ βˆ’ { 0 }
  • Bincreasing over the interval ℝ βˆ’  1 6 3 3 
  • Cincreasing over ℝ βˆ’  1 6 3 3 
  • Ddecreasing over the interval  0 , 1 6 3 3  , increasing over the intervals ] βˆ’ ∞ , 0 [ and  1 6 3 3 , ∞ 
  • Eincreasing over the interval  0 , 1 6 3 3  , decreasing over the intervals ] βˆ’ ∞ , 0 [ and  1 6 3 3 , ∞ 

Q18:

Given that 𝑓 ( π‘₯ ) = 8 π‘₯ βˆ’ 1 6 π‘₯ + 5 4 2 , determine the intervals on which 𝑓 is increasing or decreasing.

  • A 𝑓 is increasing on the intervals ] βˆ’ 1 , 0 [ and ] βˆ’ ∞ , βˆ’ 1 [ and decreasing on the intervals ] 1 , ∞ [ and ] 0 , 1 [ .
  • B 𝑓 is increasing on the intervals ] βˆ’ ∞ , βˆ’ 1 [ and ] 0 , 1 [ and decreasing on the intervals ] βˆ’ 1 , 0 [ and ] 1 , ∞ [ .
  • C 𝑓 is increasing on the intervals ] 1 , ∞ [ and ] βˆ’ ∞ , βˆ’ 1 [ and decreasing on the intervals ] βˆ’ 1 , 0 [ and ] 1 , ∞ [ .
  • D 𝑓 is increasing on the intervals ] βˆ’ 1 , 0 [ and ] 1 , ∞ [ and decreasing on the intervals ] βˆ’ ∞ , βˆ’ 1 [ and ] 0 , 1 [ .
  • E 𝑓 is increasing on the intervals ] βˆ’ 1 , 0 [ and ] 0 , 1 [ and decreasing on the intervals ] βˆ’ ∞ , βˆ’ 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 ( π‘₯ ) = 3 π‘₯ βˆ’ 9 π‘₯ βˆ’ 4 3 2 is increasing and on which it is decreasing.

  • A 𝑓 is increasing on ] βˆ’ ∞ , 0 [ and decreasing on ] 2 , ∞ [ .
  • B 𝑓 is decreasing on ] βˆ’ ∞ , 0 [ , ] 2 , ∞ [ and increasing on ] 0 , 2 [ .
  • C 𝑓 is decreasing on ] βˆ’ ∞ , 0 [ and increasing on ] 2 , ∞ [ .
  • D 𝑓 is increasing on ] βˆ’ ∞ , 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 ( π‘₯ ) = 7 π‘₯ π‘₯ + 9 2 is increasing and where it is decreasing.

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

Q24:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = 8 π‘₯ βˆ’ 7 7 π‘₯ βˆ’ 5 is increasing and where it is decreasing.

  • Adecreasing on the interval  βˆ’ ∞ , 5 7  , increasing on the interval  5 7 , ∞ 
  • Bdecreasing over ℝ βˆ’  5 7 
  • Cdecreasing over ℝ
  • Dincreasing over ℝ βˆ’  5 7 

Q25:

Determine the intervals on which the function 𝑦 = 7 π‘₯ π‘₯ βˆ’ 8 is increasing and on which it is decreasing.

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

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