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Lesson: Absolute Extrema

In this lesson, we will learn how to find the absolute maximum and minimum.

Sample Question Videos

04:40
08:45

Worksheet • 13 Questions • 2 Videos

Q1:

Determine the absolute maximum and minimum values of the function 𝑦 = βˆ’ 2 π‘₯ 3 on the interval [ βˆ’ 1 , 2 ] .

  • AThe absolute maximum is 2, and the absolute minimum is βˆ’ 1 6 .
  • BThe absolute maximum is 12, and the absolute minimum is βˆ’ 2 4 .
  • CThe absolute maximum is 6, and the absolute minimum is βˆ’ 3 2 .
  • DThe absolute maximum is 128, and the absolute minimum is βˆ’ 1 2 8 .

Q2:

Determine the absolute maximum and minimum values of the function 𝑦 = 2 π‘₯ + π‘₯ βˆ’ 3 π‘₯ βˆ’ 2 3 2 , in the interval [ βˆ’ 1 , 1 ] , approximated to two decimal places.

  • Aabsolute maximum = 0.05, absolute minimum = βˆ’ 3 . 0 2
  • Babsolute maximum = 68.00, absolute minimum = 1.00
  • Cabsolute maximum = 242.00, absolute minimum = 172.20
  • Dabsolute maximum = 9.00, absolute minimum = 1.00

Q3:

Find the absolute maximum and minimum values of the function 𝑦 = π‘₯ 4 + 1 π‘₯ βˆ’ 4 on the interval [ 1 , 3 ] .

  • AThe absolute maximum is 0, and the absolute minimum is βˆ’ 1 4 .
  • BThe absolute maximum is 5 3 6 , and the absolute minimum is βˆ’ 3 4 .
  • CThe absolute maximum is 2, and the absolute minimum is βˆ’ 3 4 .
  • DThe absolute maximum is 0, and the absolute minimum is βˆ’ 1 4 .

Q4:

Determine the absolute maximum and minimum values of the function 𝑓 ( π‘₯ ) = 2 π‘₯ βˆ’ 8 π‘₯ βˆ’ 1 3 4 2 in the interval [ βˆ’ 1 , 2 ] .

  • A The absolute maximum value is βˆ’ 1 3 , and the absolute minimum value is βˆ’ 2 1 .
  • B has no local maximum or minimum values
  • C The absolute maximum value is βˆ’ 2 1 , and the absolute minimum value is βˆ’ 1 3 .
  • D The absolute maximum value is βˆ’ 1 6 , and the absolute minimum value is 48.
  • E The absolute maximum value is 32, and the absolute minimum value is 0.

Q5:

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

  • A The absolute maximum equals 3 1 0 , and the absolute minimum equals 1 6 .
  • B The absolute maximum equals 1 1 8 , and the absolute minimum equals 1 5 0 .
  • C The absolute maximum equals 3 1 0 , and the absolute minimum equals 1 4 .
  • D The absolute maximum equals 1 4 , and the absolute minimum equals 1 6 .

Q6:

Find, if any, the local maximum and local minimum values of 𝑓 ( π‘₯ ) = 5 π‘₯ 1 3 ( π‘₯ + 1 ) 2 , together with their type.

  • Aabsolute maximum is 5 2 6 , absolute minimum is βˆ’ 5 2 6
  • B absolute maximum is 5 2 6 , absolute minimum is βˆ’ 5 2 6
  • C absolute maximum is 4 5 1 0 6 6 , absolute minimum is βˆ’ 2 5 3 3 8
  • D absolute maximum is 2 5 3 3 8 , absolute minimum is βˆ’ 4 5 1 0 6 6

Q7:

Find, if they exist, the values of the absolute maximum and/or minimum points for the function 𝑓 ( π‘₯ ) = √ 3 π‘₯ + 1 0 where π‘₯ ∈ [ βˆ’ 2 , 5 ] .

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

Q8:

Find the absolute maximum and absolute minimum of

  • A The absolute maximum value is 64 at π‘₯ = βˆ’ 1 , and the absolute minimum value is 4 at π‘₯ = 5 .
  • B The function has no absolute maximum or minimum values.
  • C The absolute maximum value is 25 at π‘₯ = βˆ’ 3 , and the absolute minimum value is 4 at π‘₯ = 5 .
  • D The absolute maximum value is 64 at π‘₯ = βˆ’ 3 , and the absolute minimum value is 25 at π‘₯ = βˆ’ 1 .
  • E The absolute maximum value is 64 at π‘₯ = βˆ’ 1 , and the absolute minimum value is 25 at π‘₯ = βˆ’ 3 .

Q9:

In the interval [ βˆ’ 1 , 2 ] , determine the absolute maximum and minimum values of the function and round them to the nearest hundredth.

  • AThe absolute maximun is 7.00, and the absolute minimum is βˆ’ 7 . 5 6 .
  • BThe absolute maximum is 11.00, and the absolute minimum is βˆ’ 4 . 1 3 .
  • CThe absolute maximum is βˆ’ 7 . 5 6 , and the absolute minimum is βˆ’ 8 . 0 0 .
  • DThe absolute maximum is 11.00, and the absolute minimum is 6.00.

Q10:

Determine the absolute maximum and minimum values of the function in the interval [ 1 , 6 ] .

  • AThe absolute maximum is 81, and the absolute minimum is βˆ’ 5 2 .
  • BThe absolute maximum is 52, and the absolute minimum is βˆ’ 9 .
  • CThe absolute maximum is 54, and the absolute minimum is 18.
  • DThe absolute maximum is 52, and the absolute minimum is 0.

Q11:

The concentration 𝐢 of a drug in a patient’s bloodstream 𝑑 hours after administration is 𝐢 ( 𝑑 ) = 1 0 0 𝑑 2 𝑑 + 7 5 2 . After about how many hours would the drug’s concentration be at its highest? If necessary, round your answer to two decimal places.

  • Aafter about 6.12 hours
  • Bafter about 8.66 hours
  • Cafter about 37.5 hours
  • Dafter about 75 hours
  • Eafter about 12.5 hours

Q12:

Find the absolute maximum and minimum values rounded to two decimal places of the function 𝑓 ( π‘₯ ) = 5 π‘₯ 𝑒 βˆ’ π‘₯ , π‘₯ ∈ [ 0 , 4 ] .

  • AThe absolute maximum is 1.84, and the absolute minimum is 0.
  • BThe absolute maximum is 1.84, and the absolute minimum is βˆ’ 1 3 . 5 9 .
  • CThe absolute maximum is 0, and the absolute minimum is 1.84.
  • DThe absolute maximum is βˆ’ 1 3 . 5 9 , and the absolute minimum is 0.
  • EThe absolute maximum is 0, and the absolute minimum is βˆ’ 1 3 . 5 9 .

Q13:

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

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