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c. i.
ii.
iii.
d. i.
ii.
iii.
5
4
3
2
1
0
2 6 x 6 3
x 6 2, x 7 3
x 2, x 3
y
x 7 2
x 6 2
x 2
y
10. f 1x2 ax 2 bx c
f ¿1x2 2ax b
Let f ¿1x2 0, then x b
2a .
If x 6 b f ¿1x2 6 0, therefore the
2a ,
function is decreasing.
If x 7 b f ¿1x2 7 0, therefore the
2a ,
function is increasing.
11. f ¿1x2 0 for x 2,
increasing: x 7 2,
decreasing: x 6 2,
local minimum: 12, 442
12.
y
8
4
–2
0
2 4
–4
5
4
3
2
13. Let y f 1x2 and u g1x2.
Let x 1 and x 2 be any two values in the
interval a x b so that x 1 6 x 2 .
Since x 1 6 x 2 , both functions are
increasing:
f 1x 2 2 7 f 1x 1 2
(1)
g1x 2 2 7 g1x 1 2
(2)
yu f 1x2 # g1x2
1
–3 –2 –1
0
1 2 3
–1
x
1 2 3 4 5
x
x
112 122 results in
f 1x 2 2 # g1x2 2 7 f 1x 1 2g1x 1 2
The function yu or f 1x2 # g1x2 is strictly
increasing.
14. strictly decreasing
a
x 1 x 1
Section 4.2, pp. 178–180
1. Determining the points on the graph of
the function for which the derivative of
the function at the x-coordinate is 0
2. a. Take the derivative of the function.
Set the derivative equal to 0. Solve
for x. Evaluate the original function
for the values of x. The (x, y) pairs
are the critical points.
b. 10, 02, 14, 322
20
y
–4
0
4 8
–20
–40
y
3. a. local minima: 12, 162, 12, 162,
local maximum: 10, 02
b. local minimum: 13, 0.32,
local maximum: (3, 0.3)
c. local minimum: 12, 52,
local maximum: (0, 1)
4. a. 10, 02, 122, 02, 122, 02
20
10
–4 –2 0 2 4
–10
b
y
f(x)
g(x)
x
x
x
b. 10, 02
c. 13, 1, 02, (0, 1)
–2
30
20
10
0
–10
–20
h(t)
0.5
–4 –2 0 2 4
–0.5
2 4 6
5. a. local minimum: (0, 3),
local maximum: (2, 27),
Tangent is parallel to the horizontal
axis for both.
b. (0, 0) neither maximum nor
minimum,
Tangent is parallel to the horizontal
axis.
c. (5, 0); neither maximum nor
minimum,
Tangent is not parallel to the
horizontal axis.
d. local minimum: 10, 12,
Tangent is parallel to the horizontal
axis.
11, 02 and (1, 0) are neither
maxima or minima.
Tangent is not parallel to the
horizontal axis for either.
6. a.
y
t
x
–20
NEL
Answers 643