3.1 Extrema on an Interval 141
3.1 Extrema on an Interval
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the value of the derivative (if it exists) of the function at the
extremum point .
a.
b.
c.
d. 0
e. 1
142 Chapter 3: Applications of Differentiation
____ 2. Find the value of the derivative (if it exists) of at the indicated
extremum.
a. is undefined.
b.
c.
d.
e.
____ 3. Find the value of the derivative (if it exists) of the function at the
extremum point .
a.
b. 0
c.
d.
e. does not exist
3.1 Extrema on an Interval 143
____ 4. Find all critical numbers of the function .
a. critical numbers:
b. critical numbers:
c. critical numbers:
d. critical numbers:
e. no critical numbers
____ 5. Find any critical numbers of the function .
a. 0
b.
c.
d.
e.
____ 6. Find all critical numbers of the function , .
a.
b.
c.
d.
e.
____ 7. Find all critical numbers of of the function , .
a.
b.
c.
d.
e.
144 Chapter 3: Applications of Differentiation
____ 8. Locate the absolute extrema of the function on the closed interval
.
a. absolute maximum:
absolute minimum:
b. absolute maximum:
absolute minimum:
c. absolute maximum:
absolute minimum:
d. absolute maximum:
absolute minimum:
e. absolute maximum:
absolute minimum:
____ 9. Locate the absolute extrema of the function on the closed
interval .
a. absolute max: ; no absolute min
b. no absolute max or min
c. no absolute max; absolute min:
d. absolute max: ; absolute min:
e. absolute max: ; absolute min:
____ 10. Locate the absolute extrema of the function on the closed interval
.
a. absolute max: ; absolute min:
b. no absolute max; absolute min:
c. absolute max: ; absolute min:
d. absolute max: ; no absolute min
e. no absolute max or min
3.1 Extrema on an Interval 145
____ 11. Locate the absolute extrema of the function on the closed interval
.
a. The absolute maximum is , and it occurs at the critical number x = 0.
The absolute minimum is , and it occurs at the left endpoint .
b. The absolute maximum is , and it occurs at either endpoint .
The absolute minimum is 0, and it occurs at the critical number x = 0.
c. The absolute maximum is , and it occurs only at the left endpoint .
The absolute minimum is 0 and it occurs at the critical number x = 0.
d. The absolute maximum is , and it occurs at the critical number x = 0.
The absolute minimum is , and it occurs at the right endpoint .
e. The absolute maximum is , and it occurs only at the right endpoint
The absolute minimum is 0 and it occurs at the critical number x = 0.
____ 12. Locate the absolute extrema of the given function on the closed interval .
a. absolute max: (0, 0); absolute min
b. absolute max: ; absolute min (0, 0)
c. absolute max: ; absolute min
d. absolute max: ; no absolute min
e. no absolute max; absolute min:
____ 13. Locate the absolute extrema of the function on the closed interval
.
a.
absolute max: (0,1) ; absolute min:
b.
absolute max: ; absolute min: (0, 1)
146 Chapter 3: Applications of Differentiation
c.
absolute max: ; no absolute min
d. no absolute max; absolute min: (0, 1)
e. absolute max: (0,1); no absolute min
____ 14. Locate the absolute extrema of the function on the closed interval
.
a. The absolute is , and it occurs at the left endpoint .
The absolute is , and it occurs at the right endpoint .
b. The absolute is , and it occurs at the right endpoint .
The absolute is , and it occurs at the left endpoint .
c. The absolute is , and it occurs at the left endpoint .
The absolute is , and it occurs at the right endpoint .
d. The absolute is , and it occurs at the right endpoint .
The absolute is and it occurs at the left endpoint .
e. The absolute is , and it occurs at the left endpoint .
The absolute is , and it occurs at the right endpoint .
____ 15. Use a graphing utility to graph the function and locate the absolute
extrema of the function on the interval .
a. absolute minimum:
no absolute maximum
b. absolute minimum:
no absolute maximum
c. absolute minimum:
no absolute maximum
d. absolute minimum:
absolute maximum:
e. absolute maximum:
absolute minimum:
3.1 Extrema on an Interval 147
____ 16. Use a graphing utility to graph the function and locate the
absolute extrema of the function on the interval .
a. left endpoint: absolute minimum
right endpoint: absolute maximum
critical points: none
b. left endpoint: absolute minimum
right endpoint: absolute maximum
critical points: none
c. left endpoint: absolute minimum
right endpoint:
critical points: absolute maximum
d. left endpoint: absolute maximum
right endpoint: absolute minimum
critical points: none
e. left endpoint: absolute minimum
right endpoint: absolute maximum
critical points:
____ 17. Use a computer algebra system to graph the function and determine all
absolute extrema on the interval .
a. absolute maximum:
absolute minimum:
b. absolute maximum:
absolute minimum:
c. absolute maximum:
absolute minimum:
d. absolute maximum:
absolute minimum:
e. absolute maximum:
absolute minimum:
148 Chapter 3: Applications of Differentiation
____ 18. The formula for the power output of battery is where is the
electromotive force in volts, is the resistance, and is the current. Find the current (measured in
amperes) that corresponds to a maximum value of P in a battery for which volts and
ohm. Assume that a -ampere fuse bounds the output in the interval . Round your answer
to two decimal places.
a. amperes
b. amperes
c. amperes
d. amperes
e. amperes
3.1 Extrema on an Interval 149
3.1 Extrema on an Interval
Answer Section
MULTIPLE CHOICE