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 = 2 𝑥 on the interval [ 1 , 2 ] .

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

Q2:

Determine the absolute maximum and minimum values of the function 𝑦 = 2 𝑥 + 𝑥 3 𝑥 2 , in the interval [ 1 , 1 ] , 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 = 3 . 0 2

Q3:

Find the absolute maximum and minimum values of the function 𝑦 = 𝑥 4 + 1 𝑥 4 on the interval [ 1 , 3 ] .

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

Q4:

Determine the absolute maximum and minimum values of the function 𝑓 ( 𝑥 ) = 2 𝑥 8 𝑥 1 3 in the interval [ 1 , 2 ] .

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

Q5:

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

  • A The absolute maximum equals 1 4 , 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 3 1 0 , and the absolute minimum equals 1 6 .

Q6:

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

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

Q7:

Find, if they exist, the values of the absolute maximum and/or minimum points for the function 𝑓 ( 𝑥 ) = 3 𝑥 + 1 0 where 𝑥 [ 2 , 5 ] .

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

Q8:

Find the absolute maximum and absolute minimum of 𝑓 ( 𝑥 ) = ( 𝑥 + 8 ) i f 3 𝑥 < 1 , 𝑥 7 i f 1 𝑥 5 .

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

Q9:

In the interval [ 1 , 2 ] , determine the absolute maximum and minimum values of the function 𝑓 ( 𝑥 ) = 4 𝑥 + 3 𝑥 7 𝑥 1 , 6 𝑥 5 𝑥 > 1 , i f i f 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 . 1 3 .
  • CThe absolute maximum is 7 . 5 6 , and the absolute minimum is 8 . 0 0 .
  • DThe absolute maximun is 7.00, and the absolute minimum is 7 . 5 6 .

Q10:

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

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

Q11:

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

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

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.

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