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Worksheet: Locating Absolute Extrema

Q1:

Determine the absolute maximum and minimum values of the function , in the interval , approximated to two decimal places.

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

Q2:

Determine the absolute maximum and minimum values of the function , in the interval , approximated to two decimal places.

  • Aabsolute maximum = 2.00, absolute minimum = 2.00
  • Babsolute maximum = 47.00, absolute minimum = 2.00
  • Cabsolute maximum = 170.80, absolute minimum = 113.10
  • Dabsolute maximum = , absolute minimum =

Q3:

Determine the absolute maximum and minimum values of the function in the interval .

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

Q4:

Determine the absolute maximum and minimum values of the function on the interval .

  • A The absolute maximum equals , and the absolute minimum equals .
  • B The absolute maximum equals , and the absolute minimum equals .
  • C The absolute maximum equals , and the absolute minimum equals .
  • D The absolute maximum equals , and the absolute minimum equals .

Q5:

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

  • A absolute maximum is 2 5 3 3 8 , absolute minimum is βˆ’ 4 5 1 0 6 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
  • Dabsolute maximum is 5 2 6 , absolute minimum is βˆ’ 5 2 6

Q6:

Find, if they exist, the values of the absolute maximum and/or minimum points for the function where .

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

Q7:

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

Q8:

In the interval , determine the absolute maximum and minimum values of the function 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 .
  • CThe absolute maximum is , and the absolute minimum is .
  • DThe absolute maximun is 7.00, and the absolute minimum is .

Q9:

Determine the absolute maximum and minimum values of the function in the interval .

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

Q10:

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

Q11:

Find the absolute maximum and minimum values rounded to two decimal places of the function , .

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

Q12:

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

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

Q13:

Consider the function 𝑓 ∢ [ 2 , 8 ) β†’ ℝ whose graph is shown.

Is it true that 𝑓 ( π‘₯ ) ≀ 5 for all π‘₯ ∈ [ 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 ( π‘₯ ) β‰₯ 2 on the interval [ 2 , 8 ) ?

  • Ano
  • Byes

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

  • Ano
  • Byes

You are told that if 𝑓 ( π‘₯ ) = 6 βˆ’ 8 π‘₯ , then 𝑓 is continuous on [ 2 , 8 ) . Why does this example not contradict the extreme value theorem?

  • Abecause the function is not a polynomial
  • Bbecause the function has a minimum
  • Cbecause the domain is bounded
  • Dbecause the domain is not a closed interval
  • Ebecause 𝑓 ( π‘₯ ) β‰₯ 0 on its domain