# 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 on the interval .

- AThe absolute maximum is 128, and the absolute minimum is .
- BThe absolute maximum is 2, and the absolute minimum is .
- CThe absolute maximum is 12, and the absolute minimum is .
- DThe absolute maximum is 6, and the absolute minimum is .

**Q2: **

Determine the absolute maximum and minimum values of the function , in the interval , 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 =
- Dabsolute maximum = 68.00, absolute minimum = 1.00

**Q3: **

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

- AThe absolute maximum is , and the absolute minimum is .
- BThe absolute maximum is 2, and the absolute minimum is .
- CThe absolute maximum is 0, and the absolute minimum is .
- DThe absolute maximum is 0, and the absolute minimum is .

**Q4: **

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

- AThe absolute maximum value is , and the absolute minimum value is 48.
- BThe absolute maximum value is , and the absolute minimum value is .
- CThe absolute maximum value is , and the absolute minimum value is .
- 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 on the interval .

- AThe absolute maximum equals , and the absolute minimum equals .
- BThe absolute maximum equals , and the absolute minimum equals .
- CThe absolute maximum equals , and the absolute minimum equals .
- DThe absolute maximum equals , and the absolute minimum equals .

**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.
- CThe function has an absolute maximum value of and an absolute minimum value of .
- DThe function has an absolute minimum value of.
- EThe function has an absolute minimum value of and an absolute maximum value of .

**Q7: **

Find the absolute maximum and absolute minimum of

- AThe absolute maximum value is 64 at , and the absolute minimum value is 25 at .
- BThe absolute maximum value is 64 at , and the absolute minimum value is 4 at .
- CThe absolute maximum value is 64 at , and the absolute minimum value is 25 at .
- DThe function has no absolute maximum or minimum values.
- EThe absolute maximum value is 25 at , and the absolute minimum value is 4 at .

**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 7.00, and the absolute minimum is .
- DThe absolute maximum is , and the absolute minimum is .

**Q9: **

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

- AThe absolute maximum is 81, and the absolute minimum is .
- BThe absolute maximum is 52, and the absolute minimum is 0.
- CThe absolute maximum is 52, and the absolute minimum is .
- 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 , .

- AThe absolute maximum is 1.84, and the absolute minimum is .
- 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 .
- EThe absolute maximum is , 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 whose graph is shown.

Is it true that for all ?

- AYes
- BNo

Does this mean that 5 is a maximum for the function on the interval ?

- ANo
- BYes

Is on the interval ?

- ANo
- BYes

Is 2 a minimum for the function on the interval ?

- 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 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 in the interval .

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