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Worksheet: First Derivative Test for Finding Local Extrema

Q1:

Find the point ( π‘₯ , 𝑓 ( π‘₯ ) ) where 𝑓 ( π‘₯ ) = | 9 π‘₯ + 9 | βˆ’ 5 has a critical point, and determine whether it is a local maximum or local minimum.

  • A ( 9 , 8 5 ) , local minimum
  • B ( βˆ’ 1 , βˆ’ 5 ) , local maximum
  • C ( 9 , 8 5 ) , local maximum
  • D ( βˆ’ 1 , βˆ’ 5 ) , local minimum
  • E ( 1 , 1 3 ) , local minimum

Q2:

Locate and classify the critical points of 𝑓 ( π‘₯ ) = 5 π‘₯ π‘₯ + 1 6 2 .

  • Alocal maximum at π‘₯ = 4 √ 3 , local minimum at π‘₯ = βˆ’ 4 √ 3
  • Blocal maximum at π‘₯ = βˆ’ 4 , local minimum at π‘₯ = 4
  • Clocal maximum at π‘₯ = βˆ’ 4 √ 3 , local minimum at π‘₯ = 4 √ 3
  • Dlocal maximum at π‘₯ = 4 , local minimum at π‘₯ = βˆ’ 4
  • Elocal maximum at π‘₯ = βˆ’ 4 , local minimum at π‘₯ = 4 √ 3

Q3:

Find the local maximum and minimum values of 𝑓 ( π‘₯ ) = βˆ’ 5 + 2 π‘₯ βˆ’ 2 π‘₯ 2 .

  • AA local maximum value is βˆ’ 1 3 2 at π‘₯ = βˆ’ 2 .
  • BA local minimum value is βˆ’ 9 2 at π‘₯ = 2 .
  • CA local minimum value is βˆ’ 1 3 2 at π‘₯ = βˆ’ 2 .
  • DA local maximum value is βˆ’ 9 2 at π‘₯ = 2 .
  • EA local maximum value is βˆ’ 9 at π‘₯ = 1 2 .

Q4:

Find the local maximum and local minimum values of 𝑓 ( π‘₯ ) = π‘₯ ( βˆ’ 6 π‘₯ + 7 ) 2 3 . Give your answers to the nearest integer.

  • AThe local minimum is 0, and the local maximum is 4.
  • BThe local maximum is 0, and the local minimum is βˆ’ 3 .
  • CThe local maximum is 0, and the local minimum is 7.
  • DThe local minimum is 0, and the local maximum is 3.

Q5:

Find (if any) the local maxima and local minima of 𝑓 ( π‘₯ ) = 2 π‘₯ √ βˆ’ π‘₯ + 2 .

  • A local maximum 4 √ 6 9 at π‘₯ = 1 .
  • B local minimum 8 √ 6 9 at π‘₯ = 4 3 .
  • C local minimum 4 √ 6 9 at π‘₯ = 1 .
  • D local maximum 8 √ 6 9 at π‘₯ = 4 3 .
  • E local maximum βˆ’ 8 √ 3 0 9 at π‘₯ = βˆ’ 4 3 .

Q6:

Determine where the local maxima and minima are for 𝑓 ( π‘₯ ) = π‘₯ 4 βˆ’ 2 π‘₯ + 5 4 2 .

  • Alocal maxima at π‘₯ = 0 , local minima at π‘₯ = 2 and π‘₯ = βˆ’ 2
  • Blocal maxima at π‘₯ = 2 and π‘₯ = βˆ’ 2 , local minima at π‘₯ = 0
  • Clocal maxima at π‘₯ = 0 , local minima at π‘₯ = 4 and π‘₯ = βˆ’ 4
  • Dlocal maxima at π‘₯ = 0 , local minima at π‘₯ = 2 and π‘₯ = βˆ’ 2

Q7:

Find, if any, the local maximum and local minimum values of 𝑓 ( π‘₯ ) = βˆ’ 2 4 π‘₯ + 1 7 π‘₯ l n .

  • AThe local minimum is 1 1 . 1 4 .
  • BThe local minimum is 2 2 . 8 6 .
  • CThe local maximum is βˆ’ 1 1 . 1 4 .
  • DThe local maximum is βˆ’ 2 2 . 8 6 .

Q8:

Identify and classify the critical points of 𝑓 ( π‘₯ ) = 2 𝑒 βˆ’ 5 π‘₯ 2 as local maxima, local minima, or neither.

  • AThere are no critical points.
  • Blocal minimum at π‘₯ = 0
  • CThe critical point at π‘₯ = 0 is neither a local maximum nor local minimum.
  • Dlocal maximum at π‘₯ = 0
  • EThis cannot be determined.

Q9:

Find, if any, the local maximum and local minimum values of 𝑓 ( π‘₯ ) = βˆ’ 5 𝑒 ( π‘₯ βˆ’ 7 ) π‘₯ .

  • Ahas no local maximum or minimum points
  • Blocal minimum value: 𝑓 ( 6 ) = 5 𝑒 6
  • Clocal maximum value: 𝑓 ( 6 ) = βˆ’ 5 𝑒 6
  • Dlocal maximum value: 𝑓 ( 6 ) = 5 𝑒 6
  • Elocal minimum value: 𝑓 ( 6 ) = βˆ’ 5 𝑒 6

Q10:

Let 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 2 4 π‘₯ c o s s i n 2 , find where the local maxima and local minima occur for 0 < π‘₯ < πœ‹ 2 .

  • Alocal maximum at π‘₯ = πœ‹ 4 , local minimum at π‘₯ = πœ‹ 2
  • Blocal maximum at π‘₯ = πœ‹ 8 , local minimum at π‘₯ = 3 πœ‹ 8
  • Clocal maximum at π‘₯ = πœ‹ 2 , local minimum at π‘₯ = πœ‹ 4
  • Dlocal maximum at π‘₯ = 3 πœ‹ 8 , local minimum at π‘₯ = πœ‹ 8
  • Elocal maximum at π‘₯ = 3 πœ‹ 8 , local minimum at π‘₯ = 2 πœ‹

Q11:

Find the point ( π‘₯ , 𝑓 ( π‘₯ ) ) where 𝑓 ( π‘₯ ) = βˆ’ | βˆ’ 4 π‘₯ | + 4 has a critical point, and determine whether it is a local maximum or local minimum.

  • A ( βˆ’ 4 , 2 0 ) , local minimum
  • B ( 0 , 4 ) , local minimum
  • C ( βˆ’ 4 , 2 0 ) , local maximum
  • D ( 0 , 4 ) , local maximum

Q12:

Find the local maximum and minimum values of 𝑓 ( π‘₯ ) = 2 π‘₯ 3 π‘₯ 2 l n .

  • Alocal maximum βˆ’ 1 3 𝑒 3 at π‘₯ = 𝑒 3 βˆ’ 3 2
  • Blocal maximum βˆ’ 1 9 𝑒 at π‘₯ = 1 3 √ 𝑒
  • Clocal minimum βˆ’ 1 3 𝑒 3 at π‘₯ = 𝑒 3 βˆ’ 3 2
  • Dlocal minimum βˆ’ 1 9 𝑒 at π‘₯ = 1 3 √ 𝑒
  • EThe function has no local maxima or minima.

Q13:

Find the critical point π‘Ž of 𝑓 ( π‘₯ ) = 4 π‘₯ + 2 4 π‘₯ c o s c o s 2 that satisfies 0 < π‘Ž < πœ‹ 2 , and then state whether ( π‘Ž , 𝑓 ( π‘Ž ) ) is a local maximum or a local minimum.

  • A π‘Ž = πœ‹ 8 , local minimum
  • B π‘Ž = πœ‹ 4 , local maximum
  • C π‘Ž = πœ‹ 8 , local maximum
  • D π‘Ž = πœ‹ 4 , local minimum