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Lesson Worksheet: Increasing and Decreasing Intervals of a Function Using Derivatives Mathematics • Higher Education

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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:

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,∞

Q2:

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,∞).

Q3:

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.

Q4:

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.

Q5:

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,∞)

Q6:

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.

Q7:

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

Q8:

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,∞)

Q9:

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.

Q10:

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,∞).

This lesson includes 72 additional questions and 470 additional question variations for subscribers.

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