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Lesson Worksheet: Absolute Extrema Mathematics • 12th Grade

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

Q1:

True or False: The extreme value theorem states that only continuous functions on closed bounded intervals have maxima and minima.

  • ATrue
  • BFalse

Q2:

Determine the absolute maximum and minimum values of the function ๐‘“(๐‘ฅ)=2๐‘ฅโˆ’8๐‘ฅโˆ’13๏Šช๏Šจ in the interval [โˆ’1,2].

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

Q3:

Find, if they exist, the values of the absolute maximum and minimum points for the function ๐‘“(๐‘ฅ)=โˆš3๐‘ฅ+10 where ๐‘ฅโˆˆ[โˆ’2,5].

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

Q4:

Determine the absolute maximum and minimum values of the function ๐‘“(๐‘ฅ)=๏ฎ(6๐‘ฅโˆ’3),๐‘ฅโ‰ค2,2โˆ’9๐‘ฅ,๐‘ฅ>2๏Šจ in the interval [1,6].

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

Q5:

Find the maximum value of the function ๐‘ฆ=โˆ’2๐‘ฅ+12๐‘ฅโˆ’25๏Šจ.

Q6:

True or False: The function ๐‘“(๐‘ฅ)=2๏€น๐‘ฅโˆ’1๏…๏Šจ๏Šฉ has an absolute minimum over the interval [โˆ’1,1] that is equal to โˆ’2.

  • ATrue
  • BFalse

Q7:

True or False: The function ๐‘“(๐‘ฅ)=2๐‘ฅ1+๐‘ฅ๏Šจ has an absolute maximum over the interval [โˆ’3,3] equal to 1.

  • AFalse
  • BTrue

Q8:

Determine the absolute maximum and minimum values of the function ๐‘ฆ=๐‘ฅ2๐‘ฅ+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.

Q9:

Find the absolute maximum and minimum values rounded to two decimal places of the function ๐‘“(๐‘ฅ)=5๐‘ฅ๐‘’๏Šฑ๏—, ๐‘ฅโˆˆ[0,4].

  • AThe absolute maximum is 1.84, and the absolute minimum is โˆ’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 โˆ’13.59.
  • EThe absolute maximum is โˆ’13.59, and the absolute minimum is 0.

Q10:

Find the absolute maximum of the function ๐‘”(๐‘ฅ)=(๐‘ฅ)2๐‘ฅln๏Šซ between ๐‘ฅ=1๐‘’ and ๐‘ฅ=๐‘’๏Šจ.

  • A52๐‘’
  • B5๐‘’2
  • C25๐‘’
  • D2๐‘’5
  • E5๐‘’

This lesson includes 40 additional questions and 278 additional question variations for subscribers.

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