Worksheet: Critical Points and Local Extrema of a Function

In this worksheet, we will practice finding critical points of a function and checking for local extrema using the first derivative test.

Q1:

Find the local maximum and minimum values of 𝑓 ( 𝑥 ) = 2 𝑥 + 3 𝑥 + 1 2 𝑥 .

  • A local maximum 7 at 𝑥 = 1 , local minimum 20 at 𝑥 = 2
  • B local maximum 13 at 𝑥 = 1 , local minimum 4 at 𝑥 = 2
  • C local maximum 4 at 𝑥 = 2 , local minimum 13 at 𝑥 = 1
  • D local maximum 20 at 𝑥 = 2 , local minimum 7 at 𝑥 = 1
  • E local maximum 8 at 𝑥 = 2 , local minimum 17 at 𝑥 = 2 0

Q2:

Determine the critical points of the function 𝑦 = 8 𝑥 in the interval [ 2 , 1 ] .

  • A ( 2 , 6 4 ) , ( 1 , 8 )
  • B ( 0 , 0 ) , ( 1 , 2 4 )
  • C ( 2 , 6 4 )
  • D ( 0 , 0 )
  • E ( 0 , 0 ) , ( 1 , 8 )

Q3:

Find the values of 𝑥 where 𝑓 ( 𝑥 ) = ( 𝑥 + 4 ) has a local maximum or a local minimum.

  • AThe function has a local minimum value at 𝑥 = 4 .
  • BThe function has a local maximum value at 𝑥 = 4 .
  • CThe function has neither local maximum nor local minimum values.
  • DThe function has a local minimum value at 𝑥 = 4 .

Q4:

Determine, if any, the local maximum and minimum values of 𝑓 ( 𝑥 ) = 2 𝑥 9 𝑥 1 2 𝑥 1 5 , together with where they occur.

  • AThe local minimum is 1 5 at 𝑥 = 0 , and there is no local maximum.
  • BThe local minimum is 2 at 𝑥 = 1 4 , and the local maximum is 1 5 at 𝑥 = 2 9 .
  • CThe local maximum is 3 8 at 𝑥 = 1 , and there is no local minimum.
  • DThe local maximum is 1 0 at 𝑥 = 1 , and the local minimum is 1 1 at 𝑥 = 2 .

Q5:

Find where (if at all) the function 𝑦 = 𝑥 1 𝑥 + 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

Q6:

Find the local maximum and local minimum values of the function 𝑓 ( 𝑥 ) = 𝑥 1 5 𝑥 1 5 𝑥 + 1 , if these exist.

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

Q7:

Determine the local minimum and local maximum values of the function 𝑦 = 9 𝑥 | 𝑥 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

Q8:

Given that the function 𝑓 ( 𝑥 ) = 𝑥 + 𝐿 𝑥 + 𝑀 has a minimum value of 2 at 𝑥 = 1 , determine the values of 𝐿 and 𝑀 .

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

Q9:

Determine the critical points of the function 𝑓 ( 𝑥 ) = 𝑥 + 6 𝑥 𝑥 0 , 𝑥 4 𝑥 𝑥 > 0 , i f i f in the interval [ 7 , 7 ] .

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

Q10:

Find the critical points of 𝑓 ( 𝑥 ) = 𝑥 ( 𝑥 1 ) .

  • AThe function has critical points at 𝑥 = 0 , 𝑥 = 1 , and 𝑥 = 1 3 .
  • BThe function has critical points at 𝑥 = 0 , 𝑥 = 2 5 , and 𝑥 = 1 .
  • CThe function has critical points at 𝑥 = 0 , 𝑥 = 1 , and 𝑥 = 1 3 .
  • DThe function has critical points at 𝑥 = 0 , 𝑥 = 2 5 , and 𝑥 = 1 .
  • EThe function has no critical points.

Q11:

Find, if any, the points ( 𝑥 , 𝑦 ) where 𝑦 = 𝑥 4 𝑥 + 2 has a local maximum or local minimum.

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

Q12:

Find the local maxima and minima of 𝑓 ( 𝑥 ) = 3 𝑥 4 2 𝑥 + 3 , if any.

  • Alocal maximum value is 1 1 3 0 at 𝑥 = 3 2
  • Blocal maximum value is 4 3 at 𝑥 = 0
  • Clocal minimum value is 1 1 3 0 at 𝑥 = 3 2
  • Dlocal minimum value is 4 3 at 𝑥 = 0
  • Elocal minimum value is 2 4 3 5 at 𝑥 = 2 3

Q13:

Find, if any, the local maxima and minima for 𝑓 ( 𝑥 ) = 3 𝑥 2 𝑥 4 𝑥 l n .

  • ALocal minimum equals 4 2 3 l n at 𝑥 = 2 3 .
  • BLocal maximum equals 1 at 𝑥 = 1 .
  • CLocal maximum equals 4 2 3 l n at 𝑥 = 2 3 .
  • DLocal minimum equals 1 at 𝑥 = 1 .
  • EIt has no local maxima and no local minima.

Q14:

Find, if any, the local maximum and local minimum values of 𝑓 ( 𝑥 ) = 𝑥 8 𝑥 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 ) = 1 . 5 5 , local maximum value
  • C 𝑓 ( 2 ) = 1 . 5 5 , local maximum value
  • D 𝑓 ( 2 ) = 1 . 5 5 , local minimum value
  • E 𝑓 ( 2 ) = 1 . 5 5 , local minimum value

Q15:

The figure shows the graph of 𝑓 ( 𝑥 ) = 5 2 ( 𝑥 ) c o s for 𝑥 > 0 .

Give an exact expression for the 𝑥 -coordinate of the point 𝑀 , including 𝜋 if necessary.

  • A 2 𝜋 ( ( 2 ) ) t a n l n
  • B 𝜋 ( ( 2 ) ) a r c t a n l n
  • C 𝜋 ( ( 2 ) ) t a n l n
  • D 2 𝜋 ( ( 2 ) ) a r c t a n l n
  • E a r c t a n l n ( ( 2 ) )

Q16:

Find (if any) the local maxima and local minima of 𝑓 ( 𝑥 ) = 2 𝑥 + 4 𝑥 + 5 .

  • Alocal maximum 8 5 + 3 2 1 5 at 𝑥 = 4 5
  • Blocal minimum 8 5 + 3 2 1 5 at 𝑥 = 4 5
  • Clocal maximum 2 + 5 at 𝑥 = 0
  • DThe function has no local maxima or minima.
  • Elocal minimum 2 + 5 at 𝑥 = 0

Q17:

Determine the value of 𝑥 where the function 𝑓 ( 𝑥 ) = 6 𝑥 has a critical point.

Q18:

Find (if any) the local maxima and local minima of 𝑓 ( 𝑥 ) = 3 𝑒 2 𝑒 + 3 .

  • Alocal maximum 3 at 𝑥 = 0
  • Blocal minimum 3 at 𝑥 = 0
  • Clocal minimum 3 𝑒 2 𝑒 + 3 at 𝑥 = 1 9
  • DThere are no local minima or maxima.
  • Elocal maximum 3 𝑒 2 𝑒 + 3 at 𝑥 = 1 9

Q19:

Find (if any) the local maxima and local minima of 𝑓 ( 𝑥 ) = 𝑒 .

  • A local maximum 𝑒 at 𝑥 = 0
  • B local minimum 1 at 𝑥 = 0
  • C local minimum 1 𝑒 at 𝑥 = 1
  • Dlocal maximum 1 at 𝑥 = 0
  • E local maximum 1 𝑒 at 𝑥 = 1

Q20:

The function ( 𝑥 ) = 𝑥 + 𝑘 𝑥 𝑒 3 2 𝑥 has a critical number at 𝑥 = 1 . Find 𝑘 and list all the critical numbers.

  • A 𝑘 = 2 , 𝑥 = 0 , 𝑥 = 1 , 𝑥 = 4
  • B 𝑘 = 2 , 𝑥 = 0 , 𝑥 = 1 , 𝑥 = 4
  • C 𝑘 = 1 , 𝑥 = 0 , 𝑥 = 1 , 𝑥 = 4
  • D 𝑘 = 2 , 𝑥 = 0 , 𝑥 = 1 , 𝑥 = 4
  • E 𝑘 = 1 , 𝑥 = 0 , 𝑥 = 1 , 𝑥 = 4

Q21:

Determine where 𝑓 ( 𝑥 ) = 3 𝑥 𝑒 has a local maximum, and give the value there.

  • A 𝑥 = 2 , 1 2 𝑒 .
  • B 𝑥 = 1 2 , 3 4 𝑒 .
  • C 𝑥 = 1 2 , 3 𝑒 4 .
  • D 𝑥 = 2 , 1 2 𝑒 .
  • E 𝑥 = 2 3 , 4 3 𝑒 .

Q22:

Determine the local maximum and minimum values of the function 𝑓 ( 𝑥 ) = 3 l n 2 𝑥 + 3 .

  • A local maximum l n 2 at 𝑥 = 1 2
  • B local minimum 3 l n 3 at 𝑥 = 0
  • C local minimum l n 2 at 𝑥 = 1 2
  • D local maximum 3 l n 3 at 𝑥 = 0
  • E local maximum l n 3 at 𝑥 = 0

Q23:

Determine, if any, the local maximum/minimum values for the function 𝑓 ( 𝑥 ) = 2 ( 𝑥 + 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

Q24:

Find (if any) the local maxima and local minima of 𝑓 ( 𝑥 ) = 𝑒 + 𝑒 6 𝑥 𝑥 .

  • Alocal maximum 3 0 7 at 𝑥 = 6 7 l n
  • Blocal maximum 6 + 6 6 7 1 7 at 𝑥 = 6 7 l n
  • Clocal minimum 3 0 7 at 𝑥 = 6 7 l n
  • Dlocal minimum 6 + 6 6 7 1 7 at 𝑥 = 6 7 l n
  • Ehas no local maxima and no local minima

Q25:

Determine the critical points of the function 𝑓 ( 𝑥 ) = 𝑥 + 1 0 𝑥 1 8 𝑥 2 , 𝑥 9 𝑥 > 2 i f i f in the interval [ 2 , 5 ] .

  • A ( 2 , 3 4 ) , ( 5 , 4 )
  • B ( 2 , 1 4 ) , ( 5 , 5 ) , ( 5 , 0 )
  • C ( 5 , 0 ) , ( 2 , 3 4 )
  • D ( 2 , 3 4 ) , ( 5 , 4 ) , ( 5 , 4 3 )
  • E ( 5 , 5 ) , ( 5 , 4 )

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