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Lesson: Local Extrema in Graphs of Functions

Worksheet • 7 Questions

Q1:

Find the maximum and minimum values of the function 𝑓 ( π‘₯ ) = 5 π‘₯ + 3 π‘₯ s i n , where 0 ≀ π‘₯ ≀ 4 πœ‹ .

  • A maximum value: 𝑓 ( 4 πœ‹ ) = 2 0 πœ‹ , minimum value: 𝑓 ( 0 ) = 0
  • B maximum value: 𝑓 ( πœ‹ ) = 5 πœ‹ , minimum value: 𝑓 ( 2 πœ‹ ) = 1 0 πœ‹
  • C maximum value: 𝑓 ( 2 πœ‹ ) = 1 0 πœ‹ , minimum value: 𝑓 ( 0 ) = 0
  • D maximum value: 𝑓 ( 2 πœ‹ ) = 1 0 πœ‹ , minimum value: 𝑓 ( πœ‹ ) = 5 πœ‹
  • E maximum value: 𝑓 ( 0 ) = 0 , minimum value: 𝑓 ( 2 πœ‹ ) = 1 0 πœ‹

Q2:

Determine the number of local extrema for the function 𝑓 ( π‘₯ ) = 7 π‘₯ + 3 π‘₯ βˆ’ 1 0 π‘₯ οŠͺ  .

Q3:

Which graph has three real zeros and two local maxima?

  • A ( ) a
  • B ( ) b
  • C ( ) c

Q4:

Using a graphing calculator to graph the function, find and classify all the extreme points of 𝑓 ( π‘₯ ) = ( π‘₯ + 2 ) ( π‘₯ βˆ’ 2 ) 2 . If necessary, round any values to 2 decimal places.

  • ALocal minimum: ( 0 . 6 7 , βˆ’ 9 . 4 8 ) , local maximum: ( βˆ’ 2 , 0 )
  • BLocal minimum: ( 0 , βˆ’ 8 ) , local maximum: ( βˆ’ 1 , 0 )
  • CLocal minimum: ( 0 , 0 ) , local maximum: ( βˆ’ 1 , 1 )
  • DLocal minimum: ( βˆ’ 2 , 0 ) , local maximum: ( 0 . 6 7 , βˆ’ 9 . 4 8 )

Q5:

Determine the local maximum and minimum values of the function 𝑦 = 2 π‘₯ + 6 π‘₯ βˆ’ 1 1 3 2 .

  • Alocal maximum value = βˆ’ 3 , local minimum value = βˆ’ 1 1
  • Blocal minimum value = βˆ’ 3
  • Clocal minimum value = βˆ’ 3 , local maximum value = βˆ’ 1 1
  • Dlocal minimum value = βˆ’ 1 1
  • Elocal maximum value = βˆ’ 3

Q6:

Find the values of the local maximum and minimum of 𝑓 ( π‘₯ ) = 3 π‘₯ + 3 π‘₯ s i n c o s given 0 ≀ π‘₯ ≀ 2 πœ‹ 3 .

  • Athe local maximum value is 𝑓 ο€» πœ‹ 1 2  = √ 2 , the local minimum value is 𝑓 ο€Ό 5 πœ‹ 1 2  = βˆ’ √ 2
  • Bthe local maximum value is 𝑓 ο€» πœ‹ 1 2  = βˆ’ √ 2 , the local minimum value is 𝑓 ο€Ό 5 πœ‹ 1 2  = √ 2
  • Cthe local maximum value is 𝑓 ο€Ό 5 πœ‹ 1 2  = βˆ’ √ 2 , the local minimum value is 𝑓 ο€» πœ‹ 1 2  = √ 2
  • Dthe local maximum value is 𝑓 ο€Ό 7 πœ‹ 1 2  = 0 , the local minimum value is 𝑓 ο€» πœ‹ 4  = 0
  • Ethe local maximum value is 𝑓 ο€» πœ‹ 4  = 0 , the local minimum value is 𝑓 ο€Ό 7 πœ‹ 1 2  = 0

Q7:

Determine the number of local extrema for the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 8 π‘₯ + 2 2 π‘₯ βˆ’ 2 4 π‘₯ + 5 4 3 2 .

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