Exam 3: Additional Applications of the Derivative

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 6 x ^ { 4 } - 48 x ^ { 3 } + 108 x ^ { 2 } + 4$

(Multiple Choice)

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Question 101

A dune buggy is on the desert at a point A located 40 km from the nearest point B on a long, straight road. The driver can travel at 45 km/hr on the desert and 75 km/hr on the road. He has an appointment in a town 28 km down the road from B in exactly one hour. He can make it.

(True/False)

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Question 102

Find the intervals of increase and decrease for $f ( x ) = \frac { 8 x - 5 } { - 2 x + 10 }$ . Round numbers to two decimal places, if necessary.

(Multiple Choice)

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Question 103

The function graphed below has a negative second derivative everywhere.

(True/False)

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Question 104

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. $f ( x ) = \frac { 2 } { x ^ { 2 } - 8 x + 7 }$

(Multiple Choice)

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Question 105

Determine where the second derivative in the function graphed below is positive.

(Short Answer)

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Question 106

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$

(Multiple Choice)

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Question 107

Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 5$ is concave up.

(Short Answer)

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Question 108

A manufacturer receives an order for 5,000 items. He owns 12 machines, each of which can produce 25 items per hour. The cost of setting up a machine for a production run is $20. Once the machines are in operation, the procedure is fully automated and can be supervised by a single worker earning $4.90 per hour. Find the number of machines that should be used in order to minimize the total cost of filling the order.

(Multiple Choice)

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Question 109

Let $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + 13$ . Find all critical points of f and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

(Short Answer)

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Question 110

Find two non-negative numbers x and y whose sum is 8 and are such that the xy² is as large as possible.

(Multiple Choice)

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Question 111

A cable is to be run from a power plant on one side of a river 600 meters wide to a factory on the other side, 2,000 meters downstream. The cost of running cable under the water is $5 per meter, while over land is $4 per meter. What is the most economical route over which to run the cable?

(Short Answer)

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Question 112

Locate all inflection points of $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ .

(Multiple Choice)

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Question 113

Find all critical numbers of the function $f ( t ) = \sqrt { t ^ { 2 } - 4 t + 8 }$ .

(Short Answer)

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Question 114

Find the absolute maximum of the function $f ( x ) = x ^ { 7 } - x ^ { 6 }$ on the interval -1 $\le$ x $\le$ 1.

(Short Answer)

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Question 115

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 8$ has

(Multiple Choice)

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Question 116

The second derivative test reveals that $f ( x ) = x ^ { 2 } - 3$ has

(Multiple Choice)

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Question 117

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval -1 $\le$ x $\le$ 3.

(Multiple Choice)

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Question 118

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3$

(Multiple Choice)

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Question 119

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$ .

(Multiple Choice)

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Question 120

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 6 x ^ { 4 } - 48 x ^ { 3 } + 108 x ^ { 2 } + 4$

A dune buggy is on the desert at a point A located 40 km from the nearest point B on a long, straight road. The driver can travel at 45 km/hr on the desert and 75 km/hr on the road. He has an appointment in a town 28 km down the road from B in exactly one hour. He can make it.

Find the intervals of increase and decrease for $f ( x ) = \frac { 8 x - 5 } { - 2 x + 10 }$ . Round numbers to two decimal places, if necessary.

The function graphed below has a negative second derivative everywhere.

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. $f ( x ) = \frac { 2 } { x ^ { 2 } - 8 x + 7 }$

Determine where the second derivative in the function graphed below is positive.

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$

Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 5$ is concave up.

Let $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + 13$ . Find all critical points of f and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Find two non-negative numbers x and y whose sum is 8 and are such that the xy² is as large as possible.

A cable is to be run from a power plant on one side of a river 600 meters wide to a factory on the other side, 2,000 meters downstream. The cost of running cable under the water is $5 per meter, while over land is $4 per meter. What is the most economical route over which to run the cable?

Locate all inflection points of $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ .

Find all critical numbers of the function $f ( t ) = \sqrt { t ^ { 2 } - 4 t + 8 }$ .

Find the absolute maximum of the function $f ( x ) = x ^ { 7 } - x ^ { 6 }$ on the interval -1 $\le$ x $\le$ 1.

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 8$ has

The second derivative test reveals that $f ( x ) = x ^ { 2 } - 3$ has

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval -1 $\le$ x $\le$ 3.

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3$

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$ .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

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Exam 3: Additional Applications of the Derivative

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. f(x)=6x4−48x3+108x2+4f ( x ) = 6 x ^ { 4 } - 48 x ^ { 3 } + 108 x ^ { 2 } + 4f(x)=6x4−48x3+108x2+4

A dune buggy is on the desert at a point A located 40 km from the nearest point B on a long, straight road. The driver can travel at 45 km/hr on the desert and 75 km/hr on the road. He has an appointment in a town 28 km down the road from B in exactly one hour. He can make it.

Find the intervals of increase and decrease for f(x)=8x−5−2x+10f ( x ) = \frac { 8 x - 5 } { - 2 x + 10 }f(x)=−2x+108x−5​ . Round numbers to two decimal places, if necessary.

The function graphed below has a negative second derivative everywhere.

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. f(x)=2x2−8x+7f ( x ) = \frac { 2 } { x ^ { 2 } - 8 x + 7 }f(x)=x2−8x+72​

Determine where the second derivative in the function graphed below is positive.

Find the intervals of increase and decrease for f(x)=6x3+9x2−108x−3f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3f(x)=6x3+9x2−108x−3

Determine where the graph of f(x)=x4−54x2+5f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 5f(x)=x4−54x2+5 is concave up.

Let f(x)=2x3−3x2−12x+13f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + 13f(x)=2x3−3x2−12x+13 . Find all critical points of f and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Find two non-negative numbers x and y whose sum is 8 and are such that the xy2 is as large as possible.

A cable is to be run from a power plant on one side of a river 600 meters wide to a factory on the other side, 2,000 meters downstream. The cost of running cable under the water is $5 per meter, while over land is $4 per meter. What is the most economical route over which to run the cable?

Locate all inflection points of f(x)=xx2+1f ( x ) = \frac { x } { x ^ { 2 } + 1 }f(x)=x2+1x​ .

Find all critical numbers of the function f(t)=t2−4t+8f ( t ) = \sqrt { t ^ { 2 } - 4 t + 8 }f(t)=t2−4t+8​ .

Find the absolute maximum of the function f(x)=x7−x6f ( x ) = x ^ { 7 } - x ^ { 6 }f(x)=x7−x6 on the interval -1 ≤\le≤ x ≤\le≤ 1.

The second derivative test reveals that f(x)=x4−2x2+8f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 8f(x)=x4−2x2+8 has

The second derivative test reveals that f(x)=x2−3f ( x ) = x ^ { 2 } - 3f(x)=x2−3 has

Find the absolute minimum of the function f(x)=x3−3x2f ( x ) = x ^ { 3 } - 3 x ^ { 2 }f(x)=x3−3x2 on the interval -1 ≤\le≤ x ≤\le≤ 3.

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. f(x)=3x4−8x3+6x2+3f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3f(x)=3x4−8x3+6x2+3

Find the intervals of increase and decrease for the function f(x)=x2+5x−3f ( x ) = x ^ { 2 } + 5 x - 3f(x)=x2+5x−3 .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 6 x ^ { 4 } - 48 x ^ { 3 } + 108 x ^ { 2 } + 4$

Find the intervals of increase and decrease for $f ( x ) = \frac { 8 x - 5 } { - 2 x + 10 }$ . Round numbers to two decimal places, if necessary.

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. $f ( x ) = \frac { 2 } { x ^ { 2 } - 8 x + 7 }$

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$

Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 5$ is concave up.

Let $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + 13$ . Find all critical points of f and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Find two non-negative numbers x and y whose sum is 8 and are such that the xy² is as large as possible.

Locate all inflection points of $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ .

Find all critical numbers of the function $f ( t ) = \sqrt { t ^ { 2 } - 4 t + 8 }$ .

Find the absolute maximum of the function $f ( x ) = x ^ { 7 } - x ^ { 6 }$ on the interval -1 $\le$ x $\le$ 1.

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 8$ has

The second derivative test reveals that $f ( x ) = x ^ { 2 } - 3$ has

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval -1 $\le$ x $\le$ 3.

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3$

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$ .