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)=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), (1,24)
  • D(2,64), (1,8)
  • E(2,64)

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)=2𝑥9𝑥12𝑥15, 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=𝑥1𝑥+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)=𝑥15𝑥15𝑥+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|𝑥3|.

  • 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.

  • A𝐿=4, 𝑀=3
  • B𝐿=2, 𝑀=3
  • C𝐿=1, 𝑀=2
  • D𝐿=2, 𝑀=1

Q9:

Determine the critical points of the function 𝑓(𝑥)=𝑥+6𝑥𝑥0,𝑥4𝑥𝑥>0,ifif in the interval [7,7].

  • A(7,49), (4,32), (0,0), (2,4), (7,21)
  • B(7,63), (4,0), (0,0), (2,0), (7,10)
  • C(4,0), (0,0), (2,0)
  • D(7,63), (0,0), (7,21)

Q10:

Find the critical points of 𝑓(𝑥)=𝑥(𝑥1).

  • 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=𝑥4𝑥+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)=3𝑥42𝑥+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)=3𝑥2𝑥4𝑥ln.

  • 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)=𝑥8𝑥ln, together with their type. Give your answers to two decimal places.

  • AThe function has no local maximum or minimum points.
  • B𝑓(2)=1.55, local minimum value
  • C𝑓(2)=1.55, local minimum value
  • D𝑓(2)=1.55, local maximum value
  • E𝑓(2)=1.55, local maximum value

Q15:

The figure shows the graph of 𝑓(𝑥)=52(𝑥)cos for 𝑥>0.

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

  • A𝜋((2))tanln
  • Barctanln((2))
  • C2𝜋((2))arctanln
  • D2𝜋((2))tanln
  • E𝜋((2))arctanln

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+3215 at 𝑥=45
  • Elocal minimum 85+3215 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)=3𝑒2𝑒+3.

  • AThere are no local minima or maxima.
  • Blocal maximum 3𝑒2𝑒+3 at 𝑥=19
  • Clocal maximum 3 at 𝑥=0
  • Dlocal minimum 3𝑒2𝑒+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 (𝑥)=𝑥+𝑘𝑥𝑒 has a critical number at 𝑥=1. Find 𝑘 and list all the critical numbers.

  • A𝑘=1, 𝑥=0, 𝑥=1, 𝑥=4
  • B𝑘=2, 𝑥=0, 𝑥=1, 𝑥=4
  • C𝑘=2, 𝑥=0, 𝑥=1, 𝑥=4
  • D𝑘=1, 𝑥=0, 𝑥=1, 𝑥=4
  • E𝑘=2, 𝑥=0, 𝑥=1, 𝑥=4

Q21:

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

  • A𝑥=12,34𝑒.
  • B𝑥=2,12𝑒.
  • C𝑥=23,43𝑒.
  • D𝑥=12,3𝑒4.
  • E𝑥=2,12𝑒.

Q22:

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

  • Alocal minimum33ln at 𝑥=0
  • Blocal maximumln3 at 𝑥=0
  • Clocal minimum2ln at 𝑥=12
  • Dlocal maximum33ln at 𝑥=0
  • Elocal maximum2ln 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+10𝑥18,𝑥2,𝑥9,𝑥>2 in the interval [2,5].

  • A(5,5), (5,4)
  • B(2,14), (5,5), (5,0)
  • C(2,34), (5,4), (5,43)
  • D(5,0), (2,34)
  • E(2,34), (5,4)

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