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Worksheet: Local Extrema

In this worksheet, we will practice finding local extreme values using the first derivative theorem.

Q1:

Find (if any) the local maxima and local minima of 𝑓 ( π‘₯ ) = 3 𝑒 βˆ’ 2 𝑒 + 3 π‘₯ π‘₯ .

  • Alocal maximum 3 at π‘₯ = 0
  • Blocal minimum 3 at π‘₯ = 0
  • Clocal minimum 3 √ 𝑒 βˆ’ 2 √ 𝑒 + 3 9 9 at π‘₯ = 1 9
  • DThere are no local minima or maxima.
  • Elocal maximum 3 √ 𝑒 βˆ’ 2 √ 𝑒 + 3 9 9 at π‘₯ = 1 9

Q2:

Find the local maxima and minima of 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 4 2 π‘₯ + 3 2 2 , if any.

  • Alocal maximum value is 1 1 3 0 at π‘₯ = 3 2
  • Blocal maximum value is βˆ’ 4 3 at π‘₯ = 0
  • Clocal minimum value is 1 1 3 0 at π‘₯ = 3 2
  • Dlocal minimum value is βˆ’ 4 3 at π‘₯ = 0
  • Elocal minimum value is βˆ’ 2 4 3 5 at π‘₯ = βˆ’ 2 3

Q3:

Find (if any) the local maxima and local minima of 𝑓 ( π‘₯ ) = 𝑒 βˆ’ π‘₯ 4 .

  • A local maximum 𝑒 at π‘₯ = 0
  • B local minimum 1 at π‘₯ = 0
  • C local minimum 1 𝑒 at π‘₯ = 1
  • Dlocal maximum 1 at π‘₯ = 0
  • E local maximum 1 𝑒 at π‘₯ = 1

Q4:

Find (if any) the local maxima and local minima of 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ + √ 4 π‘₯ + 5 2 .

  • Alocal maximum 8 5 + 3 √ 2 1 5 at π‘₯ = βˆ’ 4 5
  • Blocal minimum 8 5 + 3 √ 2 1 5 at π‘₯ = βˆ’ 4 5
  • Clocal maximum βˆ’ 2 + √ 5 at π‘₯ = 0
  • DThe function has no local maxima or minima.
  • Elocal minimum βˆ’ 2 + √ 5 at π‘₯ = 0

Q5:

Find (if any) the local maxima and local minima of 𝑓 ( π‘₯ ) = 𝑒 + 𝑒 6 π‘₯ βˆ’ π‘₯ .

  • Alocal maximum βˆ’ 3 0 7 at π‘₯ = βˆ’ 6 7 l n
  • Blocal maximum 6 + 6 βˆ’ 6 7 1 7 at π‘₯ = βˆ’ 6 7 l n
  • Clocal minimum βˆ’ 3 0 7 at π‘₯ = βˆ’ 6 7 l n
  • Dlocal minimum 6 + 6 βˆ’ 6 7 1 7 at π‘₯ = βˆ’ 6 7 l n
  • Ehas no local maxima and no local minima

Q6:

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

  • A ( 0 , βˆ’ 2 ) is a local maximum point.
  • B ( 0 , βˆ’ 2 ) is a local minimum point, and ( βˆ’ 4 , 6 ) is a local maximum point.
  • C ( βˆ’ 4 , 6 ) is a local minimum point.
  • D ( 0 , βˆ’ 2 ) is a local maximum point, and ( βˆ’ 4 , 6 ) is a local minimum point.
  • E ( 0 , βˆ’ 2 ) is a local minimum point.

Q7:

Given that the function 𝑓 ( π‘₯ ) = π‘₯ + 𝐿 π‘₯ + 𝑀 2 has a minimum value of 2 at π‘₯ = βˆ’ 1 , determine the values of 𝐿 and 𝑀 .

  • A 𝐿 = 1 , 𝑀 = 2
  • B 𝐿 = βˆ’ 2 , 𝑀 = βˆ’ 1
  • C 𝐿 = βˆ’ 4 , 𝑀 = βˆ’ 3
  • D 𝐿 = 2 , 𝑀 = 3

Q8:

Determine the local maximum and minimum values of the function 𝑓 ( π‘₯ ) = βˆ’ 3 ο€Ή 2 π‘₯ + 3  l n 2 .

  • A local maximum βˆ’ 2 l n at π‘₯ = 1 2
  • B local minimum βˆ’ 3 3 l n at π‘₯ = 0
  • C local minimum βˆ’ 2 l n at π‘₯ = 1 2
  • D local maximum βˆ’ 3 3 l n at π‘₯ = 0
  • E local maximum l n 3 at π‘₯ = 0

Q9:

Determine the local minimum and local maximum values of the function 𝑦 = 9 π‘₯ | π‘₯ βˆ’ 3 | .

  • AThe function has no local minimum or local maximum values.
  • B local minimum value = 8 1 4 , local maximum value = 0
  • C local maximum value = 8 1 4 , local minimum value = 0

Q10:

Determine the critical points of the function in the interval .

  • A ,
  • B , ,
  • C ,
  • D , ,
  • E ,

Q11:

Find, if any, the local maxima and minima for 𝑓 ( π‘₯ ) = 3 π‘₯ βˆ’ 2 π‘₯ βˆ’ 4 π‘₯ 2 l n .

  • ALocal minimum equals βˆ’ 4 ο€Ό 2 3  l n at π‘₯ = 2 3 .
  • BLocal maximum equals 1 at π‘₯ = 1 .
  • CLocal maximum equals βˆ’ 4 ο€Ό 2 3  l n at π‘₯ = 2 3 .
  • DLocal minimum equals 1 at π‘₯ = 1 .
  • EIt has no local maxima and no local minima.

Q12:

Find where (if at all) the function 𝑦 = βˆ’ π‘₯ βˆ’ 1 π‘₯ + 8 has its local maxima and minima.

  • Alocal maximum value = 6
  • Blocal minimum value = 6 , local maximum value = 1 0
  • Clocal minimum value = 1 0
  • Dlocal maximum value = 6 , local minimum value = 1 0
  • Elocal minimum value = 6

Q13:

Find the local maximum and local minimum values of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1 5 π‘₯ βˆ’ 1 5 π‘₯ + 1 2 , if these exist.

  • Alocal minimum value = βˆ’ 1 5 , no local maximum value
  • Blocal minimum value = βˆ’ 1 5 , local maximum value = βˆ’ 1 5
  • Clocal maximum value = βˆ’ 2 9 , no local minimum value
  • Dlocal maximum value = βˆ’ 1 9 , local minimum value = βˆ’ 1 5

Q14:

Determine, if any, the local maximum and minimum values of 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ βˆ’ 9 π‘₯ βˆ’ 1 2 π‘₯ βˆ’ 1 5 3 2 , together with where they occur.

  • AThe local minimum is βˆ’ 1 5 at π‘₯ = 0 , and there is no local maximum.
  • BThe local minimum is βˆ’ 2 at π‘₯ = βˆ’ 1 4 , and the local maximum is βˆ’ 1 5 at π‘₯ = 2 9 .
  • CThe local maximum is βˆ’ 3 8 at π‘₯ = 1 , and there is no local minimum.
  • DThe local maximum is βˆ’ 1 0 at π‘₯ = βˆ’ 1 , and the local minimum is βˆ’ 1 1 at π‘₯ = βˆ’ 2 .

Q15:

Find, if any, the local maximum and local minimum values of 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 8 π‘₯ 2 l n , together with their type. Give your answers to two decimal places.

  • AThe function has no local maximum or minimum points.
  • B 𝑓 ( 2 ) = βˆ’ 1 . 5 5 , local maximum value
  • C 𝑓 ( 2 ) = 1 . 5 5 , local maximum value
  • D 𝑓 ( 2 ) = βˆ’ 1 . 5 5 , local minimum value
  • E 𝑓 ( 2 ) = 1 . 5 5 , local minimum value

Q16:

Determine the value of π‘₯ where the function 𝑓 ( π‘₯ ) = 6 π‘₯ 1 8 has a critical point.

Q17:

Determine the critical points of the function in the interval [ βˆ’ 7 , 7 ] .

  • A ( βˆ’ 7 , 6 3 ) , ( 0 , 0 ) , ( 7 , 2 1 )
  • B ( βˆ’ 7 , 6 3 ) , ( βˆ’ 4 , 0 ) , ( 0 , 0 ) , ( 2 , 0 ) , ( 7 , 1 0 )
  • C ( βˆ’ 4 , 0 ) , ( 0 , 0 ) , ( 2 , 0 )
  • D ( βˆ’ 7 , βˆ’ 4 9 ) , ( βˆ’ 4 , 3 2 ) , ( 0 , 0 ) , ( 2 , βˆ’ 4 ) , ( 7 , 2 1 )

Q18:

Determine whether the following statement is true or false: A point where a function changes from an increasing to a decreasing function or vice versa is known as a turning point.

  • Atrue
  • Bfalse

Q19:

Determine, if any, the local maximum/minimum values for the function 𝑓 ( π‘₯ ) = 2 ( βˆ’ π‘₯ + 3 ) l n l n .

  • Alocal minimum value: 𝑓 ( 1 ) = 2 3 l n
  • Blocal maximum value: 𝑓 ( 1 ) = 2 3 l n
  • Clocal maximum value: 𝑓 ( 𝑒 ) = 2 2 l n
  • DThere are no local maxima/minima.
  • Elocal minimum value: 𝑓 ( 𝑒 ) = 2 2 l n

Q20:

The function β„Ž ( π‘₯ ) = ο€Ή π‘₯ + π‘˜ π‘₯  𝑒 3 2 βˆ’ π‘₯ has a critical number at π‘₯ = 1 . Find π‘˜ and list all the critical numbers.

  • A π‘˜ = βˆ’ 2 , π‘₯ = 0 , π‘₯ = 1 , π‘₯ = βˆ’ 4
  • B π‘˜ = 2 , π‘₯ = 0 , π‘₯ = 1 , π‘₯ = 4
  • C π‘˜ = βˆ’ 1 , π‘₯ = 0 , π‘₯ = 1 , π‘₯ = 4
  • D π‘˜ = βˆ’ 2 , π‘₯ = 0 , π‘₯ = 1 , π‘₯ = 4
  • E π‘˜ = βˆ’ 1 , π‘₯ = 0 , π‘₯ = 1 , π‘₯ = βˆ’ 4

Q21:

The figure shows the graph of 𝑓 ( π‘₯ ) = 5 β‹… 2 ( π‘₯ )   c o s for π‘₯ > 0 .

Give an exact expression for the π‘₯ -coordinate of the point 𝑀 , including πœ‹ if necessary.

  • A 2 πœ‹ βˆ’ ( ( 2 ) ) t a n l n
  • B πœ‹ βˆ’ ( ( 2 ) ) a r c t a n l n
  • C πœ‹ βˆ’ ( ( 2 ) ) t a n l n
  • D 2 πœ‹ βˆ’ ( ( 2 ) ) a r c t a n l n
  • E a r c t a n l n ( ( 2 ) )

Q22:

Determine where 𝑓 ( π‘₯ ) = 3 π‘₯ 𝑒 2 βˆ’ π‘₯ has a local maximum, and give the value there.

  • A π‘₯ = βˆ’ 2 , 1 2 𝑒 2 .
  • B π‘₯ = 1 2 , 3 4 √ 𝑒 .
  • C π‘₯ = βˆ’ 1 2 , 3 √ 𝑒 4 .
  • D π‘₯ = 2 , 1 2 𝑒 2 .
  • E π‘₯ = 2 3 , 4 3 𝑒 2 3 .

Q23:

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

  • A local maximum βˆ’ 7 at π‘₯ = βˆ’ 1 , local minimum 20 at π‘₯ = 2
  • B local maximum 13 at π‘₯ = 1 , local minimum 4 at π‘₯ = βˆ’ 2
  • C local maximum 4 at π‘₯ = βˆ’ 2 , local minimum 13 at π‘₯ = 1
  • D local maximum 20 at π‘₯ = 2 , local minimum βˆ’ 7 at π‘₯ = βˆ’ 1
  • E local maximum 8 at π‘₯ = 2 , local minimum 17 at π‘₯ = 2 0

Q24:

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

  • A local maximum βˆ’ 8 0 at π‘₯ = βˆ’ 4 , local minimum 28 at π‘₯ = 2
  • B local maximum βˆ’ 1 6 at π‘₯ = 4 , local minimum βˆ’ 5 2 at π‘₯ = βˆ’ 2
  • C local maximum βˆ’ 5 2 at π‘₯ = βˆ’ 2 , local minimum βˆ’ 1 6 at π‘₯ = 4
  • D local maximum 28 at π‘₯ = 2 , local minimum βˆ’ 8 0 at π‘₯ = βˆ’ 4
  • E local maximum 4 at π‘₯ = 2 , local minimum 40 at π‘₯ = 2 8

Q25:

Find the critical points of 𝑓 ( π‘₯ ) = π‘₯ ( π‘₯ βˆ’ 1 ) 2 3 .

  • AThe function has critical points at π‘₯ = 0 , π‘₯ = 1 , and π‘₯ = 1 3 .
  • BThe function has critical points at π‘₯ = 0 , π‘₯ = βˆ’ 2 5 , and π‘₯ = βˆ’ 1 .
  • CThe function has critical points at π‘₯ = 0 , π‘₯ = βˆ’ 1 , and π‘₯ = βˆ’ 1 3 .
  • DThe function has critical points at π‘₯ = 0 , π‘₯ = 2 5 , and π‘₯ = 1 .
  • EThe function has no critical points.

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