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:
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:
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.
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
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:
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.
176 Chapter 3: Applications of Differentiation
a. d.
b. e. none of the above
c.
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.
178 Chapter 3: Applications of Differentiation
3.4 Concavity and the Second Derivative Test – Answer Section