Section 4.4 Extra Practice
STUDENT BOOK PAGES 198–206
1. The following graph represents the second derivative,
f ⬙(x), of a function f (x).
2
–4 –3 –2 –1 0
–2
–4
–6
–8
–10
y
x
1
2 3 4
a. On which intervals is the graph of f (x) concave
up, and on which is the graph concave down?
b. List the x-coordinates of any points of inflection.
c. Make a rough sketch of a possible graph of f (x).
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2. Determine the critical numbers for each of the
following, and use the second derivative test to decide
if the corresponding critical point is a local maximum,
a local minimum, or neither.
a. f (x) ⫽ (x ⫺ 1)(x ⫹ 3)
b. f (x) ⫽ x 3 ⫹ 4x 2 ⫺ x ⫹ 4
15
c. f (x) ⫽ 2
x ⫺ 20
d. f (x) ⫽ x ⫺ 4x ⫺1
3. Determine the points of inflection, if they exist, for
each of the following functions.
a. f (x) ⫽ x 2 ⫹ 4x ⫺ 20
1
b. f (x) ⫽ x 3 ⫹ x 2 ⫺ 1
2
c. f (x) ⫽ 10(x ⫺ 1) 2 ⫹ x 3
1
d. f (x) ⫽ x ⫺
x
4. For the following functions, determine each critical
number and state if the corresponding critical point is
a local maximum, a local minimum, or neither. Then,
find any points of inflection.
x
a. f (x) ⫽
x⫺4
b. f (x) ⫽ x 3 ⫺ x 2 ⫹ x ⫺ 1
c. f (x) ⫽ x 4 ⫺ 4x
d. f (x) ⫽ x 4 ⫺ 100x
5. Find the value of the constant b so that the function
f (x) ⫽ x 3 ⫹ bx 2 has a point of inflection at x ⫽ ⫺1.
6. Find the value of the constant b so that the function
f (x) ⫽ x 3 ⫺ bx 2 has a point of inflection at x ⫽ 43.
7. Sketch separate graphs of a function f (x) with the
following properties:
a. f ⬘(x) ⬎ 0 for all values of x except x ⫽ 0, and
f (⫺2) ⫽ 0
b. f ⬘(x) ⬎ 0 when x ⬍ 0 and x ⬎ 2, f ⬙(1) ⫽ 0
c. f ⬘(0) ⫽ 0 and f ⬘(2) ⫽ 0
8. Use the current algorithm for curve sketching to
x 3 ⫺ 2x 2
sketch a graph of the function f (x) ⫽ x 2 ⫹ 4 .
9. Use the algorithm for curve sketching to sketch a
4
graph of the function f (x) ⫽ 兹x ⫹ 4 ⫹ x .
Section 4.4 Extra Practice
379