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 𝑓(π‘₯)=π‘₯βˆ’4π‘₯+1 is increasing and decreasing.

  • AThe function is decreasing on (1,∞), and increasing on (βˆ’βˆž,1).
  • BThe function is decreasing on (βˆ’βˆž,βˆ’1), and increasing on (βˆ’1,∞).
  • CThe function is decreasing on (βˆ’βˆž,0), and increasing on (0,∞).
  • DThe function is decreasing on (βˆ’βˆž,1), and increasing on (1,∞).
  • EThe function is decreasing on (0,∞), and increasing on (βˆ’βˆž,0).

Q2:

Given that 𝑓(π‘₯)=2π‘₯+2π‘₯sincos, 0≀π‘₯β‰€πœ‹, determine the intervals on which 𝑓 is increasing or decreasing.

  • A𝑓 is increasing on the intervals ο€Ό3πœ‹8,7πœ‹8 and ο€Ό0,3πœ‹8 and decreasing on the interval ο€Ό7πœ‹8,πœ‹οˆ.
  • B𝑓 is increasing on the interval ο€Ό7πœ‹8,πœ‹οˆ and decreasing on the intervals ο€Ό3πœ‹8,7πœ‹8 and ο€Ό0,3πœ‹8.
  • C𝑓 is increasing on the interval ο€Όπœ‹8,5πœ‹8 and decreasing on the intervals ο€»0,πœ‹8 and ο€Ό5πœ‹8,πœ‹οˆ.
  • D𝑓 is decreasing on the interval ο€Ό0,5πœ‹8 and decreasing on the interval ο€Ό5πœ‹8,πœ‹οˆ.
  • E𝑓 is increasing on the intervals ο€»0,πœ‹8 and ο€Ό5πœ‹8,πœ‹οˆ and decreasing on the interval ο€Όπœ‹8,5πœ‹8.

Q3:

Determine the intervals on which the function 𝑓(π‘₯)=(βˆ’3π‘₯βˆ’12) is increasing and on which it is decreasing.

  • Adecreasing on the interval(βˆ’βˆž,βˆ’4), increasing on the interval(βˆ’4,∞)
  • Bincreasing on the interval(βˆ’βˆž,βˆ’4), decreasing on the interval(βˆ’4,∞)
  • Cdecreasing on the intervalℝ
  • Dincreasing on the intervalℝ

Q4:

The concentration 𝐢 of a drug in a patient’s bloodstream 𝑑 hours after injection is given by 𝐢(𝑑)=2𝑑3+π‘‘οŠ¨. How does the concentration 𝐢 change as 𝑑 increases?

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

Q5:

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

  • AThe function is decreasing on (βˆ’2,0) and (0,2) and increasing on (βˆ’βˆž,βˆ’2) and (2,∞).
  • BThe function is decreasing on (βˆ’2,0) and (2,∞) and increasing on (βˆ’βˆž,βˆ’2) and (0,2).
  • CThe function is decreasing on (0,2) and (2,∞) and increasing on (βˆ’βˆž,βˆ’2) and (βˆ’2,0).
  • DThe function is decreasing on (βˆ’βˆž,βˆ’2) and (0,2) and increasing on (βˆ’2,0) and (2,∞).
  • EThe function is decreasing on (βˆ’βˆž,βˆ’2) and (βˆ’2,0) and increasing on (0,2) and (2,∞).

Q6:

Let 𝑓(π‘₯)=3π‘₯𝑒οŠͺοŠͺ. Determine the intervals where this function is increasing and where it is decreasing.

  • A𝑓 is increasing on the intervals (βˆ’βˆž,0) and (1,∞) and decreasing on the interval (0,1).
  • B𝑓 is increasing on the intervals ο€Όβˆ’βˆž,12 and ο€Ό32,∞ and decreasing on the interval ο€Ό12,32.
  • C𝑓 is decreasing on the interval (βˆ’1,0) and decreasing on the intervals (βˆ’βˆž,βˆ’1) and (0,∞).
  • D𝑓 is increasing on the interval ο€Ό32,∞ and decreasing on the intervals ο€Ό12,32 and ο€Όβˆ’βˆž,12.
  • E𝑓 is increasing on the interval (0,1) and decreasing on the intervals (βˆ’βˆž,0) and (1,∞).

Q7:

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

  • A𝑓 is decreasing on the interval ο€Ό12,∞ and decreasing on the interval ο€Ό15,12.
  • B𝑓 is increasing on the interval ο€Ό15,∞ and decreasing on the interval ο€Ό0,15.
  • C𝑓 is increasing on the interval ο€Ό0,15 and decreasing on the interval ο€Ό15,∞.
  • D𝑓 is increasing on the interval ο€Ό0,12 and decreasing on the interval ο€Ό12,∞.
  • E𝑓 is increasing on the interval ο€Ό12,∞ and decreasing on the interval ο€Ό0,12.

Q8:

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

  • AThe function is increasing on ο€½0,π‘’ο‰οŽ’οŽ‘.
  • BThe function is increasing on (0,∞).
  • CThe function is decreasing on ο€½0,π‘’ο‰οŽ’οŽ‘.
  • DThe function is decreasing on (0,∞).
  • EThe function is decreasing on 𝑒,βˆžο‰οŽ’οŽ‘.

Q9:

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

  • AThe function is increasing on 𝑒,βˆžο‰οŠ±οŽ’οŽ‘ and decreasing on ο€½0,π‘’ο‰οŠ±οŽ’οŽ‘.
  • BThe function is increasing on ο€Ώ1βˆšπ‘’,βˆžο‹ and decreasing on ο€Ώ0,1βˆšπ‘’ο‹.
  • CThe function is increasing on ο€Ίβˆšπ‘’,βˆžο† and decreasing on ο€Ί0,βˆšπ‘’ο†.
  • DThe function is increasing on ο€Ώ0,1βˆšπ‘’ο‹ and decreasing on ο€Ώ1βˆšπ‘’,βˆžο‹.
  • EThe function is increasing on ο€½0,π‘’ο‰οŠ±οŽ’οŽ‘ and decreasing on 𝑒,βˆžο‰οŠ±οŽ’οŽ‘.

Q10:

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

  • AThe function is increasing on ο€Όβˆ’βˆž,52 and decreasing on ο€Ό52,35.
  • BThe function is increasing on ο€Όβˆ’βˆž,25 and decreasing on ο€Ό25,35.
  • CThe function is increasing on ο€Ό25,35 and decreasing on ο€Όβˆ’βˆž,25.
  • DThe function is increasing on ο€Όβˆ’25,35 and decreasing on ο€Όβˆ’βˆž,βˆ’25.
  • EThe function is increasing on ο€Όβˆ’βˆž,βˆ’25 and decreasing on ο€Όβˆ’25,35.

Q11:

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

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

Q12:

Find the intervals on which the function 𝑓(π‘₯)=2π‘’βˆ’3𝑒+4 is either increasing or decreasing.

  • AThe function is increasing on (βˆ’βˆž,∞).
  • BThe function is decreasing on ο€Όβˆ’βˆž,ο€Ό43ln and ο€Όο€Ό43,∞ln.
  • CThe function is decreasing on (βˆ’βˆž,∞).
  • DThe function is increasing on ο€Όβˆ’βˆž,ο€Ό43ln and ο€Όο€Ό43,∞ln.
  • EThe function is increasing on ο€Όβˆ’βˆž,ο€Ό34ln and ο€Όο€Ό34,∞ln.

Q13:

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

  • Aincreasing over ℝ
  • Bdecreasing over β„βˆ’{βˆ’3}
  • Cincreasing over β„βˆ’{βˆ’3}
  • Dincreasing over β„βˆ’{3}
  • Eincreasing over (βˆ’βˆž,βˆ’3), decreasing over (βˆ’3,∞)

Q14:

Determine the intervals over which the function 𝑓(π‘₯)=βˆ’|2π‘₯|+28 is increasing and over which it is decreasing.

  • Aincreasing overβ„βˆ’{14}
  • Bdecreasing overβ„βˆ’{14}
  • Cdecreasing over the interval (βˆ’βˆž,0), increasing over the interval (0,∞)
  • Ddecreasing over the interval (0,∞), increasing over the interval (βˆ’βˆž,0)
  • Edecreasing over the interval (14,∞), increasing over the interval (βˆ’βˆž,14)

Q15:

Let 𝑓(π‘₯)=π‘₯βˆ’19π‘₯≀1,2π‘₯βˆ’20π‘₯>1.ifif Find the intervals on which 𝑓 is increasing and where it is decreasing.

  • Aincreasing over the interval (βˆ’βˆž,0), decreasing over the interval (0,∞)
  • Bdecreasing over β„βˆ’{1}
  • Cincreasing over the interval (βˆ’βˆž,1), decreasing over the interval (1,∞)
  • Ddecreasing over the interval (βˆ’βˆž,0), increasing over the interval (0,∞)
  • Eincreasing over β„βˆ’{1}

Q16:

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

  • Aincreasing over ℝ
  • Bincreasing over the intervals ο€Όβˆ’βˆž,βˆ’1027 and (0,∞), decreasing over the interval ο€Όβˆ’1027,0
  • Cdecreasing over ℝ
  • Dincreasing over the interval ο€Όβˆ’1027,0, decreasing over the intervals ο€Όβˆ’βˆž,βˆ’1027 and (0,∞)

Q17:

Determine the intervals over which the function 𝑓(π‘₯)=11π‘₯βˆ’8π‘₯ is increasing and over which it is decreasing.

  • Aincreasing over the interval ο€Ό0,1633, decreasing over the intervals (βˆ’βˆž,0) and 1633,∞
  • Bincreasing over the interval β„βˆ’ο¬1633
  • Cincreasing over β„βˆ’ο¬1633
  • Dincreasing over β„βˆ’{0}
  • Edecreasing over the interval ο€Ό0,1633, increasing over the intervals (βˆ’βˆž,0) and ο€Ό1633,∞

Q18:

Given that 𝑓(π‘₯)=8π‘₯βˆ’16π‘₯+5οŠͺ, determine the intervals on which 𝑓 is increasing or decreasing.

  • A𝑓 is increasing on the intervals (βˆ’βˆž,βˆ’1) and (0,1) and decreasing on the intervals (βˆ’1,0) and (1,∞).
  • B𝑓 is increasing on the intervals (βˆ’1,0) and (0,1) and decreasing on the intervals (βˆ’βˆž,βˆ’1) and (1,∞).
  • C𝑓 is increasing on the intervals (βˆ’1,0) and (βˆ’βˆž,βˆ’1) and decreasing on the intervals (1,∞) and (0,1).
  • D𝑓 is increasing on the intervals (1,∞) and (βˆ’βˆž,βˆ’1) and decreasing on the intervals (βˆ’1,0) and (1,∞).
  • E𝑓 is increasing on the intervals (βˆ’1,0) and (1,∞) and decreasing on the intervals (βˆ’βˆž,βˆ’1) and (0,1).

Q19:

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

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

Q20:

Determine the intervals on which the function 𝑓(π‘₯)=3π‘₯βˆ’9π‘₯βˆ’4 is increasing and on which it is decreasing.

  • A𝑓 is decreasing on (βˆ’βˆž,0),(2,∞) and increasing on (0,2).
  • B𝑓 is increasing on (βˆ’βˆž,0),(2,∞) and decreasing on (0,2).
  • C𝑓 is decreasing on (βˆ’βˆž,0) and increasing on (2,∞).
  • D𝑓 is increasing on (βˆ’βˆž,0) and decreasing on (2,∞).

Q21:

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

  • AThe function is increasing on (2πœ‹,4πœ‹) and decreasing on (0,2πœ‹).
  • BThe function is increasing on (0,πœ‹) and decreasing on (πœ‹,4πœ‹).
  • CThe function is increasing on (0,2πœ‹) and decreasing on (2πœ‹,4πœ‹).
  • DThe function is increasing on (0,4πœ‹).
  • EThe function is decreasing on (0,4πœ‹).

Q22:

For 0<π‘₯<2πœ‹5, find the intervals on which 𝑓(π‘₯)=5π‘₯+35π‘₯coscos is increasing or decreasing.

  • AThe function is decreasing on ο€»0,πœ‹10 and increasing on ο€Όπœ‹10,2πœ‹5.
  • BThe function is decreasing on ο€Όπœ‹10,2πœ‹5 and increasing on ο€»0,πœ‹10.
  • CThe function is decreasing on ο€Όπœ‹5,2πœ‹5 and increasing on ο€»0,πœ‹5.
  • DThe function is decreasing on ο€»0,πœ‹5 and increasing on ο€Όπœ‹5,2πœ‹5.
  • EThe function is decreasing on ο€»0,πœ‹5 and increasing on ο€Όπœ‹10,2πœ‹5.

Q23:

Determine the intervals on which the function 𝑓(π‘₯)=7π‘₯π‘₯+9 is increasing and where it is decreasing.

  • Aincreasing on the intervals (βˆ’βˆž,βˆ’3) and (3,∞), decreasing on the interval (βˆ’3,3)
  • Bincreasing on the interval (βˆ’βˆž,βˆ’3), decreasing on the interval (3,∞)
  • Cdecreasing on the intervals (βˆ’βˆž,βˆ’3) and (3,∞), increasing on the interval (βˆ’3,3)
  • Ddecreasing on the interval (βˆ’βˆž,βˆ’3), increasing on the interval (3,∞)

Q24:

Determine the intervals on which the function 𝑓(π‘₯)=8π‘₯βˆ’77π‘₯βˆ’5 is increasing and where it is decreasing.

  • Aincreasing over β„βˆ’ο¬57
  • Bdecreasing over β„βˆ’ο¬57
  • Cdecreasing on the interval ο€Όβˆ’βˆž,57, increasing on the interval ο€Ό57,∞
  • Ddecreasing over ℝ

Q25:

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

  • Adecreasing on ℝ
  • Bdecreasing on β„βˆ’{8}
  • Cdecreasing on (βˆ’βˆž,8), increasing on (8,∞)
  • Dincreasing on ℝ
  • Edecreasing on (8,∞), increasing on (βˆ’βˆž,8)

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