Worksheet: Second Derivative Test for Local Extrema

In this worksheet, we will practice classifying local extrema using the second derivative test.

Q1:

Find, if any, the points (π‘₯,𝑦) where 𝑦=π‘₯+3π‘₯βˆ’16 has a local maximum or local minimum.

  • A ( βˆ’ 2 , βˆ’ 1 2 ) is the local minimum point, and the function does not have a local maximum point.
  • B ( βˆ’ 2 , βˆ’ 1 2 ) is the local maximum point, and (0,βˆ’16)is the local minimum point.
  • C ( 0 , βˆ’ 1 6 ) is the local minimum point, and the function does not have a local maximum point.
  • D ( βˆ’ 2 , βˆ’ 1 2 ) is the local maximum point, and the function does not have a local minimum point.
  • E ( βˆ’ 2 , βˆ’ 1 2 ) is the local minimum point, and (0,βˆ’16)is the local maximum point.

Q2:

Find, if any, the point (π‘₯,𝑦) where 𝑦=βˆ’π‘₯+4π‘₯βˆ’6 has a local maximum or local minimum.

  • A ( βˆ’ 2 , βˆ’ 1 8 ) is a local maximum point.
  • B ( βˆ’ 2 , βˆ’ 1 8 ) is a local minimum point.
  • C ( 2 , βˆ’ 2 ) is a local minimum point.
  • DThe function does not have local maximum or minimum points.
  • E ( 2 , βˆ’ 2 ) is a local maximum point.

Q3:

Find the local maxima/minima of the function 𝑓(π‘₯)=3π‘₯βˆ’2π‘₯οŠͺ.

  • A ο€Ό βˆ’ 1 2 , 7 1 6  is a local maximum point.
  • B ο€Ό 1 2 , βˆ’ 1 1 6  is a local minimum point.
  • C ο€Ό βˆ’ 1 2 , 7 1 6  is a local minimum point.
  • D ο€Ό 1 2 , βˆ’ 1 1 6  is a local maximum point.
  • EThe function does not have local maximum or minimum points.

Q4:

Use the second derivative test to find, if possible, the local maximum and minimum values of 𝑓(π‘₯)=9π‘₯βˆ’2π‘₯βˆ’5οŠͺ.

  • Alocal minimum value =βˆ’5
  • Blocal maximum value =βˆ’469
  • Clocal minimum value =βˆ’469, local maximum value =βˆ’5
  • Dlocal minimum value =βˆ’5, local maximum value =βˆ’469

Q5:

Determine the local maximum and minimum values of the function 𝑦=βˆ’3π‘₯βˆ’6π‘₯βˆ’4.

  • Alocal minimum value=βˆ’1
  • Blocal maximum value=βˆ’13
  • Clocal minimum value=βˆ’13
  • DIt has no local maximum or minimum values.
  • Elocal maximum value=βˆ’1

Q6:

Determine the local maximum and local minimum values of 𝑓(π‘₯)=4π‘₯βˆ’12π‘₯βˆ’5.

  • Alocal maximum value βˆ’5 at π‘₯=√3, local minimum value βˆ’5 at π‘₯=βˆ’βˆš3
  • Blocal maximum value 8 at π‘₯=βˆ’1, local minimum value 8 at π‘₯=1
  • Clocal maximum value βˆ’13 at π‘₯=1, local minimum value 3 at π‘₯=βˆ’1
  • Dlocal maximum value 8 at π‘₯=1, local minimum value 8 at π‘₯=βˆ’1
  • Elocal maximum value 3 at π‘₯=βˆ’1, local minimum value βˆ’13 at π‘₯=1

Q7:

Find where (if at all) the function 𝑓(π‘₯)=βˆ’2π‘₯βˆ’9π‘₯βˆ’12π‘₯βˆ’15 has its local maxima and minima.

  • Alocal minimum at π‘₯=βˆ’1, no local maximum
  • Blocal maximum at π‘₯=βˆ’2, local minimum at π‘₯=βˆ’1
  • Clocal maximum at π‘₯=βˆ’1, local minimum at π‘₯=βˆ’2
  • Dlocal minimum at π‘₯=βˆ’14, local maximum at π‘₯=29

Q8:

Find the points (π‘₯,𝑦) where 𝑦=9π‘₯+9π‘₯ has a local maximum or a local minimum.

  • A ( βˆ’ 1 , βˆ’ 1 8 ) is a local maximum point.
  • B ( 1 , 1 8 ) is a local maximum point and (βˆ’1,βˆ’18) is a local minimum point.
  • C ( 1 , 1 8 ) is a local minimum point.
  • D ( 1 , 1 8 ) is a local minimum point and (βˆ’1,βˆ’18) is a local maximum point.
  • EThe function doesn’t have local maximum or minimum points.

Q9:

Find, if any, the local maximum and local minimum values of 𝑓(π‘₯)=19π‘₯+15π‘₯sincos, together with their type.

  • Aabsolute maximum = 23.04, absolute minimum = βˆ’19.73
  • Babsolute maximum = βˆ’24.21, absolute minimum = 24.21
  • Cabsolute maximum = 24.21, absolute minimum = βˆ’24.21
  • Dabsolute maximum = βˆ’19.73, absolute minimum = 23.04

Q10:

Find, if any, the local maximum and local minimum values of 𝑦=7π‘₯+7π‘₯.

  • AThe function does not have local maximum or minimum points.
  • BThe local maximum value isβˆ’14.
  • CThe local maximum value is14, and the local minimum value isβˆ’14.
  • DThe local minimum value is14, and the local maximum value isβˆ’14.
  • EThe local minimum value is14.

Q11:

Find the local maxima and local minima of 𝑓(π‘₯)=βˆ’5π‘₯3+2π‘₯βˆ’16π‘₯ln, if any.

  • Alocal minimum 712βˆ’16ο€Ό12ln at π‘₯=12 , local maximum 1160βˆ’16ο€Ό110ln at π‘₯=110
  • Blocal minimum 815βˆ’16ο€Ό25ln at π‘₯=25 , local maximum βˆ’83βˆ’162ln at π‘₯=2
  • Clocal minimum 13βˆ’16ο€Ό15ln at π‘₯=15 , local maximum 13 at π‘₯=13
  • Dlocal minimum 13 at π‘₯=1 , local maximum 13βˆ’16ο€Ό15ln at π‘₯=15
  • Elocal minimum 1160βˆ’16ο€Ό110ln at π‘₯=110 , local maximum 712βˆ’16ο€Ό12ln at π‘₯=12

Q12:

Find the local maximum and minimum values of 𝑓(π‘₯)=2√π‘₯βˆ’4√π‘₯.

  • Alocal minimum 0 at π‘₯=16
  • Blocal maximum 0 at π‘₯=16
  • Clocal minimum βˆ’2 at π‘₯=1
  • Dno local maximum and no local minimum values
  • Elocal maximum βˆ’2 at π‘₯=1

Q13:

Suppose 𝑓′(4)=0 and 𝑓′′(4)=βˆ’4. What can you say about 𝑓 at point π‘₯=4?

  • A 𝑓 has a vertical tangent at π‘₯=4.
  • BIt is not possible to state the nature of the turning point of 𝑓 at π‘₯=4.
  • C 𝑓 has a local minimum at π‘₯=4.
  • D 𝑓 has an inflection point at π‘₯=4.
  • E 𝑓 has a local maximum at π‘₯=4.

Q14:

Find, if any, the local maximum and local minimum values of 𝑓(π‘₯)=5π‘₯13(π‘₯+1), together with their type.

  • Aabsolute maximum is 25338, absolute minimum is βˆ’451,066
  • Babsolute maximum is 526, absolute minimum is βˆ’526
  • Cabsolute maximum is 526, absolute minimum is βˆ’526
  • Dabsolute maximum is 451,066, absolute minimum is βˆ’25338

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