Worksheet: Absolute Extrema

In this worksheet, we will practice finding the absolute maximum and minimum values of a function over a given interval using derivatives.

Q1:

Determine the absolute maximum and minimum values of the function 𝑦=2𝑥 on the interval [1,2].

  • AThe absolute maximum is 128, and the absolute minimum is 128.
  • BThe absolute maximum is 2, and the absolute minimum is 16.
  • CThe absolute maximum is 12, and the absolute minimum is 24.
  • DThe absolute maximum is 6, and the absolute minimum is 32.

Q2:

Determine the absolute maximum and minimum values of the function 𝑦=2𝑥+𝑥3𝑥2, in the interval [1,1], approximated to two decimal places.

  • Aabsolute maximum = 242.00, absolute minimum = 172.20
  • Babsolute maximum = 9.00, absolute minimum = 1.00
  • Cabsolute maximum = 0.05, absolute minimum = 3.02
  • Dabsolute maximum = 68.00, absolute minimum = 1.00

Q3:

Find the absolute maximum and minimum values of the function 𝑦=𝑥4+1𝑥4 on the interval [1,3].

  • AThe absolute maximum is 536, and the absolute minimum is 34.
  • BThe absolute maximum is 2, and the absolute minimum is 34.
  • CThe absolute maximum is 0, and the absolute minimum is 14.
  • DThe absolute maximum is 0, and the absolute minimum is 14.

Q4:

Determine the absolute maximum and minimum values of the function 𝑓(𝑥)=2𝑥8𝑥13 in the interval [1,2].

  • AThe absolute maximum value is 16, and the absolute minimum value is 48.
  • BThe absolute maximum value is 21, and the absolute minimum value is 13.
  • CThe absolute maximum value is 13, and the absolute minimum value is 21.
  • DThe absolute maximum value is 32, and the absolute minimum value is 0.
  • Ehas no local maximum or minimum values

Q5:

Determine the absolute maximum and minimum values of the function 𝑦=𝑥2𝑥+8 on the interval [2,6].

  • AThe absolute maximum equals 14, and the absolute minimum equals 16.
  • BThe absolute maximum equals 310, and the absolute minimum equals 14.
  • CThe absolute maximum equals 118, and the absolute minimum equals 150.
  • DThe absolute maximum equals 310, and the absolute minimum equals 16.

Q6:

Find, if they exist, the values of the absolute maximum and/or minimum points for the function 𝑓(𝑥)=3𝑥+10 where 𝑥[2,5].

  • AThe function has no absolute maximum or minimum points.
  • BThe function has an absolute maximum value of5.
  • CThe function has an absolute maximum value of 2and an absolute minimum value of 5.
  • DThe function has an absolute minimum value of2.
  • EThe function has an absolute minimum value of 2and an absolute maximum value of 5.

Q7:

Find the absolute maximum and absolute minimum of 𝑓(𝑥)=(𝑥+8),3𝑥<1,𝑥7,1𝑥5.

  • AThe absolute maximum value is 64 at 𝑥=1, and the absolute minimum value is 25 at 𝑥=3.
  • BThe absolute maximum value is 64 at 𝑥=1, and the absolute minimum value is 4 at 𝑥=5.
  • CThe absolute maximum value is 64 at 𝑥=3, and the absolute minimum value is 25 at 𝑥=1.
  • DThe function has no absolute maximum or minimum values.
  • EThe absolute maximum value is 25 at 𝑥=3, and the absolute minimum value is 4 at 𝑥=5.

Q8:

In the interval [1,2], determine the absolute maximum and minimum values of the function 𝑓(𝑥)=4𝑥+3𝑥7𝑥1,6𝑥5𝑥>1,ifif and round them to the nearest hundredth.

  • AThe absolute maximum is 11.00, and the absolute minimum is 6.00.
  • BThe absolute maximum is 11.00, and the absolute minimum is 4.13.
  • CThe absolute maximum is 7.00, and the absolute minimum is 7.56.
  • DThe absolute maximum is 7.56, and the absolute minimum is 8.00.

Q9:

Determine the absolute maximum and minimum values of the function 𝑓(𝑥)=(6𝑥3),𝑥2,29𝑥,𝑥>2 in the interval [1,6].

  • AThe absolute maximum is 81, and the absolute minimum is 52.
  • BThe absolute maximum is 52, and the absolute minimum is 0.
  • CThe absolute maximum is 52, and the absolute minimum is 9.
  • DThe absolute maximum is 54, and the absolute minimum is 18.

Q10:

Find the absolute maximum and minimum values rounded to two decimal places of the function 𝑓(𝑥)=5𝑥𝑒, 𝑥[0,4].

  • AThe absolute maximum is 1.84, and the absolute minimum is 13.59.
  • BThe absolute maximum is 1.84, and the absolute minimum is 0.
  • CThe absolute maximum is 0, and the absolute minimum is 1.84.
  • DThe absolute maximum is 0, and the absolute minimum is 13.59.
  • EThe absolute maximum is 13.59, and the absolute minimum is 0.

Q11:

If a continuous function on an interval is bounded below but does not achieve a minimum, what can we conclude?

  • AThe interval is not bounded.
  • BThe interval is not closed and it is not bounded.
  • CThe interval is the entire number line.
  • DThe interval is not closed.
  • EEither the interval is not closed or it is not bounded.

Q12:

Consider the function 𝑓[2,8) whose graph is shown.

Is it true that 𝑓(𝑥)5 for all 𝑥[2,8)?

  • AYes
  • BNo

Does this mean that 5 is a maximum for the function 𝑓 on the interval [2,8)?

  • ANo
  • BYes

Is 𝑓(𝑥)2 on the interval [2,8)?

  • ANo
  • BYes

Is 2 a minimum for the function 𝑓 on the interval [2,8)?

  • AYes
  • BNo

Why does this example not contradict the extreme value theorem?

  • ABecause the function has a minimum
  • BBecause the domain is not a closed interval
  • CBecause 𝑓(𝑥)0 on its domain
  • DBecause the function is not a polynomial
  • EBecause the domain is bounded

Q13:

Find the absolute maximum and minimum values of the function 𝑓(𝑥)=𝑥+3𝑥,𝑥0,𝑥6𝑥,𝑥>0 in the interval [5,13].

  • AThe absolute maximum equals 4, and the absolute minimum equals 9.
  • BThe absolute maximum equals 45, and the absolute minimum equals 20.
  • CThe absolute maximum equals 91, and the absolute minimum equals 9.
  • DThe absolute maximum equals 91, and the absolute minimum equals 50.
  • EThe absolute maximum equals 4, and the absolute minimum equals 50.

Q14:

Find the absolute maximum of the function 𝑔(𝑥)=(𝑥)2𝑥ln between 𝑥=1𝑒 and 𝑥=𝑒.

  • A52𝑒
  • B5𝑒2
  • C25𝑒
  • D2𝑒5
  • E5𝑒

Q15:

Consider the function 𝑓(𝑥)=(𝑥3)𝑥5,4𝑥16𝑥>5,ifif over the interval [0,7].

Determine the absolute minimum of 𝑓(𝑥) over the given interval.

Determine the absolute maximum of 𝑓(𝑥) over the given interval.

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