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Worksheet: Second Derivative Test for Local Extrema

Q1:

Find, if any, the points ( π‘₯ , 𝑦 ) where 𝑦 = π‘₯ + 3 π‘₯ βˆ’ 1 6 3 2 has a local maximum or local minimum.

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

Q2:

Use the second derivative test to find, if possible, the local maximum and minimum values of 𝑓 ( π‘₯ ) = 9 π‘₯ βˆ’ 2 π‘₯ βˆ’ 5 4 2 .

  • A local maximum value = βˆ’ 4 6 9
  • B local minimum value = βˆ’ 5 , local maximum value = βˆ’ 4 6 9
  • C local minimum value = βˆ’ 5
  • D local minimum value = βˆ’ 4 6 9 , local maximum value = βˆ’ 5

Q3:

Find the local maxima and local minima of 𝑓 ( π‘₯ ) = βˆ’ 5 π‘₯ 3 + 2 π‘₯ βˆ’ 1 6 π‘₯ 2 l n , if any.

  • Alocal minimum 1 3 βˆ’ 1 6 ο€Ό 1 5  l n at π‘₯ = 1 5 , local maximum 1 3 at π‘₯ = 1 3
  • Blocal minimum 7 1 2 βˆ’ 1 6 ο€Ό 1 2  l n at π‘₯ = 1 2 , local maximum 1 1 6 0 βˆ’ 1 6 ο€Ό 1 1 0  l n at π‘₯ = 1 1 0
  • Clocal minimum 1 3 at π‘₯ = 1 , local maximum 1 3 βˆ’ 1 6 ο€Ό 1 5  l n at π‘₯ = 1 5
  • D local minimum 1 1 6 0 βˆ’ 1 6 ο€Ό 1 1 0  l n at π‘₯ = 1 1 0 , local maximum 7 1 2 βˆ’ 1 6 ο€Ό 1 2  l n at π‘₯ = 1 2
  • Elocal minimum 8 1 5 βˆ’ 1 6 ο€Ό 2 5  l n at π‘₯ = 2 5 , local maximum βˆ’ 8 3 βˆ’ 1 6 2 l n at π‘₯ = 2

Q4:

Find the local maxima/minima of the function 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 2 π‘₯ 4 3 .

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

Q5:

Determine the local maximum and minimum values of the function 𝑦 = βˆ’ 3 π‘₯ βˆ’ 6 π‘₯ βˆ’ 4 2 .

  • Alocal maximum value = βˆ’ 1 3
  • Blocal minimum value = βˆ’ 1
  • Clocal minimum value = βˆ’ 1 3
  • Dlocal maximum value = βˆ’ 1
  • E It has no local maximum or minimum values.

Q6:

Determine the local maximum and local minimum values of 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 1 2 π‘₯ βˆ’ 5 3 .

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

Q7:

Find the local maximum and minimum values of the curve that passes through the point ( βˆ’ 1 , 7 ) where the gradient of the tangent is 6 ο€Ή π‘₯ + 4 π‘₯ + 3  2 .

  • AThe local maximum value is 7, and the local minimum value is βˆ’ 1 .
  • BThe local maximum value is βˆ’ 5 , and the local minimum value is βˆ’ 1 3 .
  • CThe local maximum value is 7, and the local minimum value is 15.
  • DThe local maximum value is 15, and the local minimum value is 7.

Q8:

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

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

Q9:

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

  • Alocal maximum 0 at π‘₯ = 1 6
  • Blocal maximum βˆ’ 2 at π‘₯ = 1
  • Clocal minimum 0 at π‘₯ = 1 6
  • Dlocal minimum βˆ’ 2 at π‘₯ = 1
  • Eno local maximum and no local minimum values

Q10:

Find, if any, the point ( π‘₯ , 𝑦 ) where 𝑦 = βˆ’ π‘₯ + 4 π‘₯ βˆ’ 6 2 has a local maximum or local minimum.

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

Q11:

Find, if any, the local maximum and local minimum values of 𝑓 ( π‘₯ ) = 1 9 π‘₯ + 1 5 π‘₯ s i n c o s , together with their type.

  • Aabsolute maximum = 23.04, absolute minimum = βˆ’ 1 9 . 7 3
  • Babsolute maximum = βˆ’ 2 4 . 2 1 , absolute minimum = 24.21
  • Cabsolute maximum = βˆ’ 1 9 . 7 3 , absolute minimum = 23.04
  • Dabsolute maximum = 24.21, absolute minimum = βˆ’ 2 4 . 2 1

Q12:

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

  • A The local minimum value is 1 4 .
  • B The local maximum value is 1 4 , and the local minimum value is βˆ’ 1 4 .
  • C The local maximum value is βˆ’ 1 4 .
  • D The local minimum value is 1 4 , and the local maximum value is βˆ’ 1 4 .
  • EThe function does not have local maximum or minimum points.

Q13:

Find where (if at all) the function 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ βˆ’ 9 π‘₯ βˆ’ 1 2 π‘₯ βˆ’ 1 5 3 2 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 minimum at π‘₯ = βˆ’ 1 4 , local maximum at π‘₯ = 2 9
  • Dlocal maximum at π‘₯ = βˆ’ 1 , local minimum at π‘₯ = βˆ’ 2