Chapter 3 Use The Second Relative Min Relative Max

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170 Chapter 3: Applications of Differentiation
3.4 Concavity and the Second Derivative Test
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Determine the open intervals on which the graph of is concave
downward or concave upward.
a. concave downward on
b. concave upward on ; concave downward on
c. concave upward on ; concave downward on
d. concave upward on
e. concave downward on ; concave upward on
____ 2. Determine the open intervals on which the graph of is
concave downward or concave upward.
a. concave downward on
b. concave upward on ; concave downward on
c. concave upward on ; concave downward on
d. concave downward on ; concave upward on
e. concave downward on ; concave upward on
____ 3. Determine the open intervals on which the graph of the function is
concave upward or concave downward.
a.
concave upward: ; concave downward:
b.
concave upward: ; concave downward:
c. concave upward: ; concave downward:
d. concave upward: ; concave downward:
e. concave upward: ; concave downward:
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3.4 Concavity and the Second Derivative Test 171
____ 4. Determine the open intervals on which the graph of is concave
downward or concave upward.
a. concave downward on ; concave upward on
b. concave downward on ; concave upward on
c. concave upward on ; concave downward on
d. concave downward on ; concave upward on
e. concave upward on ; concave downward on
____ 5. Determine the open intervals on which the graph of the function
is concave upward or concave downward.
a. concave upward: ; concave downward:
b. concave upward: ; concave downward:
c. concave upward: ; concave downward:
d. concave upward: ; concave downward:
e. concave upward: ; concave downward:
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172 Chapter 3: Applications of Differentiation
____ 6. Find all points of inflection on the graph of the function .
a.
b.
c.
d.
e.
____ 7. Find the points of inflection and discuss the concavity of the function.
a. inflection point at ; concave upward on ; concave downward on
b. inflection point at ; concare downward on ; concave upward on
c. inflection point at ; concave downward on ; concave upward on
d. inflection point at ; concave downward on ; concave upward on
e. inflection point at ; concave upward on ; concave downward on
____ 8. Find the points of inflection and discuss the concavity of the function
.
a. no inflection points; concave up on
b. no inflection points; concave down on
c. inflection point at x = 16; concave up on
d. inflection point at x = 0; concave up on ; concave down on
e. inflection point at x = 16; concave down on
____ 9. Find all points of inflection, if any exist, of the graph of the function
. Round your answers to two decimal places.
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3.4 Concavity and the Second Derivative Test 173
a.
b.
c.
d.
e. no points of inflection
____ 10. Find the point of inflection of the graph of the function on the interval
.
a.
b.
c.
d.
e.
____ 11. Find the points of inflection and discuss the concavity of the function
on the interval .
a. no inflection points. concave up on
b. concave upward on ; concave downward on ; inflection point at
c. no inflection points. concave down on
d. concave downward on ; concave upward on ; inflection point at
e. none of the above
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174 Chapter 3: Applications of Differentiation
____ 12. Find all points of inflection of the graph of the function on
the interval . Round your answer to three decimal places wherever applicable.
a.
b.
c.
d.
e.
____ 13. Find the points of inflection and discuss the concavity of the function
on the interval .
a. concave down on ; no points of inflection
b. concave downward on ; concave upward on ; inflection
points at and
c. concave upward on ; concave downward on ; inflection
points at and
d. concave up on ; no points of inflection
e. none of the above
____ 14. Find all relative extrema of the function . Use the Second
Derivative Test where applicable.
a. relative max: ; no relative min
b. no relative max; no relative min
c. relative min: ; relative max:
d. relative min: ; no relative max
e. relative min: ; relative max:
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3.4 Concavity and the Second Derivative Test 175
____ 15. Find all relative extrema of the function . Use the Second
Derivative Test where applicable.
a. relative max: ; no relative min
b. relative min: ; no relative max
c. relative min: ; no relative max
d. relative max: ; no relative min
e. no relative max or min
____ 16. Find all relative extrema of the function . Use the Second Derivative
Test where applicable.
a. relative minimum:
b. relative minimum:
c. relative maximum:
d. relative minimum:
e. relative maximum:
____ 17. Find all relative extrema of the function . Use the Second Derivative
Test where applicable.
a. relative max: (1, –2); no relative min
b. no relative max or min
c. relative min: (0, –3); no relative max
d. relative max: (1, –2); relative min: (0, –3)
e. relative max: (0, –3); no relative min
____ 18. The graph of f is shown. Graph f, f' and f'' on the same set of coordinate axes.
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176 Chapter 3: Applications of Differentiation
a. d.
b. e. none of the above
c.
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3.4 Concavity and the Second Derivative Test 177
____ 19. Find the cubic function of the form where and the
coefficients are real numbers, which satisfies the conditions given below.
Relative maximum:
Relative minimum:
Inflection point:
a.
b.
c.
d.
e.
____ 20. Suppose a manufacturer has determined that the total cost C of operating a
factory is where x is the number of units produced. At what level of
production will the average cost per unit be minimized? (The average cost per unit is C/x.)
a.
b.
c.
d.
e.
____ 21. Suppose the annual sales S of a new product is given by where
t is time in years. Find the exact time when the annual sales are increasing at the greatest rate. Round
your answer to three decimal places.
a.
b.
c.
d.
e.
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178 Chapter 3: Applications of Differentiation
3.4 Concavity and the Second Derivative Test - Answer Section

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