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Worksheet: Increasing and Decreasing Intervals of a Function Using Derivatives

Q1:

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 ο€Ό βˆ’ ∞ , ο€Ό 4 3   l n and ο€Ό ο€Ό 4 3  , ∞  l n .
  • CThe function is decreasing on ( βˆ’ ∞ , ∞ ) .
  • DThe function is increasing on ο€Ό βˆ’ ∞ , ο€Ό 4 3   l n and ο€Ό ο€Ό 4 3  , ∞  l n .
  • EThe function is increasing on ο€Ό βˆ’ ∞ , ο€Ό 3 4   l n and ο€Ό ο€Ό 3 4  , ∞  l n .

Q2:

Find all possible intervals on which the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ + 1 2 2 is increasing and decreasing.

  • A The function is decreasing on ( βˆ’ ∞ , 1 ) , and increasing on ( 1 , ∞ ) .
  • B The function is decreasing on ( 0 , ∞ ) , and increasing on ( βˆ’ ∞ , 0 ) .
  • C The function is decreasing on ( 1 , ∞ ) , and increasing on ( βˆ’ ∞ , 1 ) .
  • D The function is decreasing on ( βˆ’ ∞ , 0 ) , and increasing on ( 0 , ∞ ) .
  • E The function is decreasing on ( βˆ’ ∞ , βˆ’ 1 ) , and increasing on ( βˆ’ 1 , ∞ ) .

Q3:

Find the intervals on which the function 𝑓 ( π‘₯ ) = 4 π‘₯ π‘₯ 2 l n is increasing and decreasing.

  • AThe function is increasing on ο€Ί √ 𝑒 , ∞  and decreasing on ο€Ί 0 , √ 𝑒  .
  • BThe function is increasing on ο€Ώ 0 , 1 √ 𝑒  and decreasing on ο€Ώ 1 √ 𝑒 , ∞  .
  • CThe function is increasing on ο€½ 𝑒 , ∞  βˆ’ 3 2 and decreasing on ο€½ 0 , 𝑒  βˆ’ 3 2 .
  • DThe function is increasing on ο€Ώ 1 √ 𝑒 , ∞  and decreasing on ο€Ώ 0 , 1 √ 𝑒  .
  • EThe function is increasing on ο€½ 0 , 𝑒  βˆ’ 3 2 and decreasing on ο€½ 𝑒 , ∞  βˆ’ 3 2 .

Q4:

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

  • Aincreasing on the interval
  • Bincreasing on the interval , decreasing on the interval
  • Cdecreasing on the interval
  • Ddecreasing on the interval , increasing on the interval

Q5:

Find the intervals on which the function 𝑓 ( π‘₯ ) = 5 π‘₯ √ βˆ’ 5 π‘₯ + 3 is increasing and decreasing.

  • A The function is increasing on ο€Ό βˆ’ ∞ , βˆ’ 2 5  and decreasing on ο€Ό βˆ’ 2 5 , 3 5  .
  • B The function is increasing on ο€Ό 2 5 , 3 5  and decreasing on ο€Ό βˆ’ ∞ , 2 5  .
  • C The function is increasing on ο€Ό βˆ’ 2 5 , 3 5  and decreasing on ο€Ό βˆ’ ∞ , βˆ’ 2 5  .
  • D The function is increasing on ο€Ό βˆ’ ∞ , 2 5  and decreasing on ο€Ό 2 5 , 3 5  .
  • E The function is increasing on ο€Ό βˆ’ ∞ , 5 2  and decreasing on ο€Ό 5 2 , 3 5  .

Q6:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = βˆ’ 1 + 1 π‘₯ βˆ’ 4 π‘₯ 2 is increasing and decreasing.

  • AThe function is decreasing on ( βˆ’ ∞ , 0 ) and ( 0 , 8 ) and increasing on ( 8 , ∞ ) .
  • BThe function is decreasing on ( 0 , 8 ) and increasing on ( βˆ’ ∞ , 0 ) and ( 8 , ∞ ) .
  • CThe function is decreasing on ( 8 , ∞ ) and increasing on ( βˆ’ ∞ , 0 ) and ( 0 , 8 ) .
  • D The function is decreasing on ( βˆ’ ∞ , 0 ) and ( 8 , ∞ ) and increasing on ( 0 , 8 ) .
  • EThe function is decreasing on ( βˆ’ ∞ , 0 ) and increasing on ( 8 , ∞ ) and ( 0 , 8 ) .

Q7:

Determine the intervals on which the function 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ π‘₯ s i n , where 0 ≀ π‘₯ ≀ 4 πœ‹ , is increasing and where it is decreasing.

  • A The function is increasing on ( 0 , 2 πœ‹ ) and decreasing on ( 2 πœ‹ , 4 πœ‹ ) .
  • B The function is decreasing on ( 0 , 4 πœ‹ ) .
  • CThe function is increasing on ( 2 πœ‹ , 4 πœ‹ ) and decreasing on ( 0 , 2 πœ‹ ) .
  • D The function is increasing on ( 0 , 4 πœ‹ ) .
  • E The function is increasing on ( 0 , πœ‹ ) and decreasing on ( πœ‹ , 4 πœ‹ ) .

Q8:

Given that 𝑓 ( π‘₯ ) = 5 π‘₯ βˆ’ 3 π‘₯ βˆ’ π‘₯ 2 l n , find the intervals on which 𝑓 is increasing or decreasing.

  • A 𝑓 is increasing on the interval ο€Ό 1 5 , ∞  and decreasing on the interval ο€Ό 0 , 1 5  .
  • B 𝑓 is increasing on the interval ο€Ό 0 , 1 2  and decreasing on the interval ο€Ό 1 2 , ∞  .
  • C 𝑓 is increasing on the interval ο€Ό 0 , 1 5  and decreasing on the interval ο€Ό 1 5 , ∞  .
  • D 𝑓 is increasing on the interval ο€Ό 1 2 , ∞  and decreasing on the interval ο€Ό 0 , 1 2  .
  • E 𝑓 is decreasing on the interval ο€Ό 1 2 , ∞  and decreasing on the interval ο€Ό 1 5 , 1 2  .

Q9:

For 0 < π‘₯ < 2 πœ‹ 5 , find the intervals on which 𝑓 ( π‘₯ ) = 5 π‘₯ + 3 5 π‘₯ c o s c o s 2 is increasing or decreasing.

  • A The function is decreasing on ο€» 0 , πœ‹ 1 0  and increasing on ο€Ό πœ‹ 1 0 , 2 πœ‹ 5  .
  • B The function is decreasing on ο€Ό πœ‹ 5 , 2 πœ‹ 5  and increasing on ο€» 0 , πœ‹ 5  .
  • C The function is decreasing on ο€Ό πœ‹ 1 0 , 2 πœ‹ 5  and increasing on ο€» 0 , πœ‹ 1 0  .
  • D The function is decreasing on ο€» 0 , πœ‹ 5  and increasing on ο€Ό πœ‹ 5 , 2 πœ‹ 5  .
  • E The function is decreasing on ο€» 0 , πœ‹ 5  and increasing on ο€Ό πœ‹ 1 0 , 2 πœ‹ 5  .

Q10:

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

  • Aincreasing over the interval , decreasing over the interval
  • Bincreasing over the interval , decreasing over the interval
  • Cincreasing over
  • Ddecreasing over the interval , increasing over the interval
  • Edecreasing over

Q11:

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

  • Aincreasing on the interval , decreasing on the interval
  • Bincreasing on the intervals and , decreasing on the interval
  • Cdecreasing on the interval , increasing on the interval
  • Ddecreasing on the intervals and , increasing on the interval

Q12:

Find the intervals on which the function 𝑓 ( π‘₯ ) = 5 ( βˆ’ 4 π‘₯ + 6 ) l n l n is increasing and decreasing.

  • A The function is decreasing on ( 0 , ∞ ) .
  • BThe function is increasing on ο€½ 0 , 𝑒  3 2 .
  • CThe function is increasing on ( 0 , ∞ ) .
  • D The function is decreasing on ο€½ 0 , 𝑒  3 2 .
  • E The function is decreasing on ο€½ 𝑒 , ∞  3 2 .

Q13:

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

  • A increasing over
  • B increasing over the interval , decreasing over the intervals and
  • C decreasing over
  • D increasing over the intervals and , decreasing over the interval

Q14:

Determine the intervals over which the function is increasing and over which it is decreasing.

  • Aincreasing over
  • Bincreasing over the interval
  • Cincreasing over
  • Ddecreasing over the interval , increasing over the intervals and
  • Eincreasing over the interval , decreasing over the intervals and

Q15:

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 , ∞ ) .

Q16:

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 ο€Ό 3 πœ‹ 8 , 7 πœ‹ 8  and ο€Ό 0 , 3 πœ‹ 8  and decreasing on the interval ο€Ό 7 πœ‹ 8 , πœ‹  .
  • B 𝑓 is increasing on the interval ο€Ό πœ‹ 8 , 5 πœ‹ 8  and decreasing on the intervals ο€» 0 , πœ‹ 8  and ο€Ό 5 πœ‹ 8 , πœ‹  .
  • C 𝑓 is increasing on the interval ο€Ό 7 πœ‹ 8 , πœ‹  and decreasing on the intervals ο€Ό 3 πœ‹ 8 , 7 πœ‹ 8  and ο€Ό 0 , 3 πœ‹ 8  .
  • D 𝑓 is increasing on the intervals ο€» 0 , πœ‹ 8  and ο€Ό 5 πœ‹ 8 , πœ‹  and decreasing on the interval ο€Ό πœ‹ 8 , 5 πœ‹ 8  .
  • E 𝑓 is decreasing on the interval ο€Ό 0 , 5 πœ‹ 8  and decreasing on the interval ο€Ό 5 πœ‹ 8 , πœ‹  .

Q17:

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

  • AThe function is increasing on ( βˆ’ ∞ , βˆ’ 1 ) and ( βˆ’ 1 , 1 ) and decreasing on ( 1 , ∞ ) .
  • BThe function is increasing on ( βˆ’ 1 , 1 ) and decreasing on ( βˆ’ ∞ , βˆ’ 1 ) and ( 1 , ∞ ) .
  • CThe function is increasing on ( 1 , ∞ ) and decreasing on ( βˆ’ ∞ , βˆ’ 1 ) and ( βˆ’ 1 , 1 ) .
  • DThe function is increasing on ( βˆ’ ∞ , βˆ’ 1 ) and ( 1 , ∞ ) and decreasing on ( βˆ’ 1 , 1 ) .
  • EThe function is increasing on ( βˆ’ ∞ , βˆ’ 1 ) and decreasing on ( βˆ’ 1 , 1 ) and ( 1 , ∞ ) .

Q18:

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 ο€Ό βˆ’ ∞ , 1 2  and ο€Ό 3 2 , ∞  and decreasing on the interval ο€Ό 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 ο€Ό 3 2 , ∞  and decreasing on the intervals ο€Ό 1 2 , 3 2  and ο€Ό 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 , ∞ ) .

Q19:

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.

Q20:

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

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

Q21:

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

  • Adecreasing on the interval , increasing on the interval
  • Bdecreasing over
  • Cdecreasing over
  • Dincreasing over

Q22:

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

  • A decreasing on , increasing on
  • B decreasing on , increasing on
  • C decreasing on
  • D decreasing on
  • E increasing on

Q23:

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

  • Aincreasing over
  • Bincreasing over
  • Cdecreasing over
  • Dincreasing over
  • Eincreasing over , decreasing over

Q24:

Determine the intervals over which the function is increasing and over which it is decreasing.

  • Aincreasing over
  • Bdecreasing over the interval , increasing over the interval
  • Cdecreasing over
  • D decreasing over the interval , increasing over the interval
  • Edecreasing over the interval , increasing over the interval

Q25:

Determine the intervals on which 𝑓 ( π‘₯ ) = π‘₯ 2 βˆ’ 4 π‘₯ + 2 4 2 is increasing or decreasing.

  • A The function is decreasing on ( βˆ’ ∞ , βˆ’ 2 ) and ( βˆ’ 2 , 0 ) and increasing on ( 0 , 2 ) and ( 2 , ∞ ) .
  • B The function is decreasing on ( βˆ’ 2 , 0 ) and ( 2 , ∞ ) and increasing on ( βˆ’ ∞ , βˆ’ 2 ) and ( 0 , 2 ) .
  • C The function is decreasing on ( 0 , 2 ) and ( 2 , ∞ ) and increasing on ( βˆ’ ∞ , βˆ’ 2 ) and ( βˆ’ 2 , 0 ) .
  • D The function is decreasing on ( βˆ’ ∞ , βˆ’ 2 ) and ( 0 , 2 ) and increasing on ( βˆ’ 2 , 0 ) and ( 2 , ∞ ) .
  • E The function is decreasing on ( βˆ’ 2 , 0 ) and ( 0 , 2 ) and increasing on ( βˆ’ ∞ , βˆ’ 2 ) and ( 2 , ∞ ) .