Worksheet: Critical Points and Local Extrema of a Function

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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)=−2𝑥+3𝑥+12𝑥.

  • Alocal maximum −7 at 𝑥=−1, local minimum 20 at 𝑥=2
  • Blocal maximum 20 at 𝑥=2, local minimum −7 at 𝑥=−1
  • Clocal maximum 13 at 𝑥=1, local minimum 4 at 𝑥=−2
  • Dlocal maximum 8 at 𝑥=2, local minimum 17 at 𝑥=20
  • Elocal maximum 4 at 𝑥=−2, local minimum 13 at 𝑥=1

Q2:

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

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

Q3:

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

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

Q4:

Determine, if any, the local maximum and minimum values of 𝑓(𝑥)=−2𝑥−9𝑥−12𝑥−15, together with where they occur.

  • AThe local maximum is −38 at 𝑥=1, and there is no local minimum.
  • BThe local maximum is −10 at 𝑥=−1, and the local minimum is −11 at 𝑥=−2.
  • CThe local minimum is −15 at 𝑥=0, and there is no local maximum.
  • DThe local minimum is −2 at 𝑥=−14, and the local maximum is −15 at 𝑥=29.

Q5:

Determine (if there are any) the values of the local maximum and the local minimum of the function 𝑦=−𝑥−1𝑥+8.

  • ALocal maximum value=6, local minimum value=10
  • BLocal maximum value=6
  • Clocal minimum value=10
  • DLocal minimum value=6
  • ELocal minimum value=6, local maximum value=10

Q6:

Find the local maximum and local minimum values of the function 𝑓(𝑥)=𝑥−15𝑥−15𝑥+1, if these exist.

  • Alocal maximum value =−19, local minimum value =−15
  • Blocal minimum value =−15, no local maximum value
  • Clocal minimum value =−15, local maximum value =−15
  • Dlocal maximum value =−29, no local minimum value

Q7:

Determine the local minimum and local maximum values of the function 𝑦=9𝑥|𝑥−3|.

  • AThe function has no local minimum or local maximum values.
  • Blocal minimum value =814, local maximum value =0
  • Clocal maximum value =814, local minimum value =0

Q8:

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

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

Q9:

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

  • A ( − 7 , − 4 9 ) , ( − 4 , 3 2 ) , ( 0 , 0 ) , ( 2 , − 4 ) , ( 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 , 6 3 ) , ( 0 , 0 ) , ( 7 , 2 1 )

Q10:

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

  • AThe function has critical points at 𝑥=0, 𝑥=−25, and 𝑥=−1.
  • BThe function has critical points at 𝑥=0, 𝑥=1, and 𝑥=13.
  • CThe function has no critical points.
  • DThe function has critical points at 𝑥=0, 𝑥=25, and 𝑥=1.
  • EThe function has critical points at 𝑥=0, 𝑥=−1, and 𝑥=−13.

Q11:

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

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

Q12:

Find the local maxima and minima of 𝑓(𝑥)=3𝑥−42𝑥+3, if any.

  • Alocal minimum value is −43 at 𝑥=0
  • Blocal maximum value is −43 at 𝑥=0
  • Clocal maximum value is 1130 at 𝑥=32
  • Dlocal minimum value is 1130 at 𝑥=32
  • Elocal minimum value is −2435 at 𝑥=−23

Q13:

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

  • ALocal maximum equals 1 at 𝑥=1.
  • BLocal minimum equals −423ln at 𝑥=23.
  • CIt has no local maxima and no local minima.
  • DLocal minimum equals 1 at 𝑥=1.
  • ELocal maximum equals −423ln at 𝑥=23.

Q14:

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

Q15:

The figure shows the graph of 𝑓(𝑥)=5⋅2(𝑥)cos for 𝑥>0.

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

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

Q16:

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

  • AThe function has no local maxima or minima.
  • Blocal maximum −2+√5 at 𝑥=0
  • Clocal minimum −2+√5 at 𝑥=0
  • Dlocal maximum 85+3√215 at 𝑥=−45
  • Elocal minimum 85+3√215 at 𝑥=−45

Q17:

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

Q18:

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

  • AThere are no local minima or maxima.
  • Blocal maximum 3√𝑒−2√𝑒+3 at 𝑥=19
  • Clocal maximum 3 at 𝑥=0
  • Dlocal minimum 3√𝑒−2√𝑒+3 at 𝑥=19
  • Elocal minimum 3 at 𝑥=0

Q19:

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

  • Alocal maximum 1𝑒 at 𝑥=1
  • Blocal maximum 𝑒 at 𝑥=0
  • Clocal minimum 1𝑒 at 𝑥=1
  • Dlocal minimum 1 at 𝑥=0
  • Elocal maximum 1 at 𝑥=0

Q20:

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

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

Q21:

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

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

Q22:

Determine the local maximum and minimum values of the function 𝑓(𝑥)=−32𝑥+3ln.

  • Alocal minimum−33ln at 𝑥=0
  • Blocal maximumln3 at 𝑥=0
  • Clocal minimum−2ln at 𝑥=12
  • Dlocal maximum−33ln at 𝑥=0
  • Elocal maximum−2ln at 𝑥=12

Q23:

Determine, if any, the local maximum/minimum values for the function 𝑓(𝑥)=2(−𝑥+3)lnln.

  • Alocal maximum value: 𝑓(𝑒)=22ln
  • Blocal maximum value: 𝑓(1)=23ln
  • Clocal minimum value: 𝑓(𝑒)=22ln
  • Dlocal minimum value: 𝑓(1)=23ln
  • EThere are no local maxima/minima.

Q24:

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

  • Alocal maximum 6+6 at 𝑥=−67ln
  • Blocal minimum −307 at 𝑥=−67ln
  • Clocal minimum 6+6 at 𝑥=−67ln
  • Dlocal maximum −307 at 𝑥=−67ln
  • Ehas no local maxima and no local minima

Q25:

Determine the critical points of the function 𝑓(𝑥)=𝑥+10𝑥−18𝑥≤2,𝑥−9𝑥>2ifif in the interval [−2,5].

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

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