160
—3
—1
y
3
F(x)=x’°+2
\lnnecrion
poini
!
-x
2t
3x + 3
2 +
t
—
3
0
2
2. g(x)
4. f(x)
8. f(x)
2 + 12x 6. f(t)
9x
6,O
2
cos
=
(x
=
4
3x
3
4x
—
—
—
32
=
1
—
(x + 1)(x
—
2
3 -I- 3t
t
3
s
=
sl
=
=
22. F(x)
6x + 12
3 + 2
4x
24. H(s)
—
3 + 1
5x
=
=
=
—
1
—
12
x
2
2 -I- cos
2x
—
3.4
2 + 1
x
2)
=
’
2
3x
F”(.v)
=
=
9x’
Find all points of inflection of F( x)
F(s)
f()
f”(x)
[(0)
=
=
23(1
1;f(6)
—
=
=
x)
3 -I- 2.
.r’
28. g(x)
=
8x1’3 +
3; increasing and concave down on
0;f(6)
Don (0, 3); f’ (x)
3;f(3)
<
3;f(2)
=
2;f(6)
=
0 on (3, 6);
4;
>
=
0;
=
0;
> Don (0, 5);f”(x) < Don (5, 6)
=
=
=
0; f’(s)
f’(r)
3;f(2)
=
2;f(6)
4;f(4)
=
0;
2;f(6)
>
=
0;
Don (3,4)
—ion (5, 6);
<0 on (2,4) U (4, 5);
=
2); f’ (t) <0 on (2,4) U (4, 6);
f’ (x) > 0 on (0,1) U (3, 4);
0;
1;f(2)
Don (0,1) U (2,6);f’(x) > Don (1,2)
f’(.x) < Oon (0, 2) U (2, 6);f’(2)
<
f’(4)
Don (0,
[(4)
>
=
f’(x)
=
=
Don (1, 3) U (4,6)
f(3)
<
=
f’(4)
f”(x) < Don (0,3) U (4,5);f”(x)
[‘(2)
f’(x) > Don (0, 2);
34. [(0)
f”(x)
f’(2)
33. [(0)
32.
31. f(0)
8;f(6) = —2; decreasing on (0, 6); inflection
30. f(0)
point at the ordered pair (2.3), concave up on (2.6)
29. f(0)
(0, 6)
[0, 6] that satisfies all the stated conditions.
In Problems 29—34, sketch the graph of a continuous fnnction f on
27.
The second derivative, F” (x), is never 0; however, it fails to exist at x = 0. The
point (0,2) is an inflection point since F”(x) > 0 for x < 0 and F”(x) < 0 for
x > 0. The graph is sketched in Figure 19.
SOLUTION
• EXAMPLE 7
Chapter 3 Applications of the Derivative
—2
Figure 19
Concepts
Problem Set 3.2
=
=
3
2x
=—
=
—
In Problems 1—10, use the Mon otonicity Theorem to find where
the given function is increasing and where it is decreasing.
1. f(x)
3. h(t)
5. G(x)
7. h(z)
=
9. H(t)=sint,0t2ir
10. R(0)
12. G(w)
11. f(x)
17. F(s)
—
In Problems 11—18, use the Concavity Theorem to determine
where the given function is concave up and where it is concave
down. Also find all inflection points.
1)2
14. f(z)
3
3t
—
18t
=
—
—
13. T(t)
4
x
2
2 -I- 3x + 1
24x
=
3
4 + 8x
x
—
15. q(x)
=
2x
2 + 12 sin
24x
3
6x
16. f(x)
=
—
18. G(x)
20. g(x)
=
5
3x
=
In Problems 19—28, determine where the graph of the given fluic
tion is increasing, decreasing, concave up, and concave down. Then
sketch the graph (see Example 4).
21. g(x)
=
19. f(x)=x
—12x-l-1
3
23. G(x)
35. Prove that a quadratic function has no point of inflection.
36. Prove that a cubic function has exactly one point of
inflection.
37. Prove that, if f’(x) exists and is continuous on an interval
I and if f’(x)
0 at all interior points of I, then either f is
—
jncreasing throughout I or decreasing throughout I. 1-lint: Use the
Intermediate Value Theorem to show that there cannot be two
2 of I where has opposite signs.
1 and x
f’
points x
is
38. Suppose that f is a function whose derivative
2 ± 1). Use Problem 37 to prove thatf
x H- 1)/(.v
2
(x
f(s)
is jncreasiflg everywhere.
39. Use the Wlonotonicity Theorem to prove each statement
iffl < x < y.
(c)
<
(b)
2
(a) x2 < y
,
.
—
—
—
49. What conditions on a, b, and c will make f(s) =
2 + cx + d always increasing?
3 + bx
as
b/sLi: has the
41. Determine a and b so that f(x) = a/ Hpoint (4,13) as an inflection point.
42. Suppose that the cubic function f(s) has three real zeros,
. Show that its inflection point has s-coordinate
3
, and r
2
r
).
3
)(x r
2
r
) (x
1
)/3. 1-lint: f(s) = a(x r
3
2 -I- r
1 -I- r
(r
43. Suppose that f(s) > 0 and g’(x) > 0 for all x. What
simple additional conditions (if any) are needed to guarantee
that:
(a) f(s) + g(x) is increasing for alIx;
(b) f(s) g(x) is increasing for all x;
(c) f(g(x)) is increasing for all x?
44. Suppose that f(s) > 0 and g”(r) > 0 for all x. What
simple additional conditions (if an’) are needed to guarantee
that
(a) f(s) -I- g(.r) is concave up for aIls;
(b) f(s) .g(x)is concave up for alIx:
(c) f(g(s)) is concave up for ails?
Use a graphing calculator or a computer to do Problems
45-48.
f’(x)
=
—
52 -F-
=
2 onl
46. Repeat Problem 45forf(x)
=
[—2, 41. Where on lisf
(x/3) on (0,10).
2
xcos
(—2,7).
45. Letf(x) = sinx -t- cos(s/2) on the intervall
(a) Draw the graph offon I.
(b) Use this graph to estimate where f’(x) < OonI.
(c) Use this graph to estimate where f”(x) < Don 1.
(d) Plot the graph off’ to confirm your answer to part (b).
(e) Plot the graph of f” to confirm your answer to part (c).
47. Let
increasing?
—
48. Let f”(x) =
5x H- 42 4 on I = [—2, 31. Where
on Iisf concave down?
49. Translate each of the following into the language of deriv
atives of distance with respect to time.For each part, sketch a plot
of the car’s position s against time t, and indicate the concavity.
(a) The speed of the car is proportional to the distance it has
traveled.
(b) The car is speeding up.
(c) I didn’t say the car was slowing down; I said its rate of in
crease in speed was slowing down.
(d) The car’s speed is increasing 10 miles per hour every minute.
(e)
The
car
is slowing very gently to a stop.
(f)
The
car always travels the same distance in equal time
intervals.
Section 3.2 Monotonicity and Concavity
161
50. Translate each of the following into the language of deriv
atives, sketch a plot of the appropriate function and indicate the
concavity.
(a) Water is evaporating from the tank at a constant rate.
(b) Water is being poured into the tank at 3 gallons per minute
but is also leaking out at gallon per minute.
(c) Since water is being poured into the conical tank at a con
stant rate, the water level is rising at a slower and slower
rate.
(d) Inflation held steady this ‘ear but is expected to rise more
and more rapidly in the years ahead.
(e) At present the price of oil is dropping. but this trend is ex
pected to slow and then reverse direction in 2 years.
(f) David’s temperature is still rising, but the penicillin seems to
be taking effect.
51. Translate each of the following statements into mathe
matical language, sketch a plot of the appropriate function, and
indicate the concavity.
(a) The cost of a car continues to increase and at a faster and
faster rate.
(b) During the last 2 years, the United States has continued to
cut its consumption of oil, but at a slower and slower rate.
(c) World population continues to grow, but at a slower and
slower rate.
(d) The angle that the Leaning Tower of Pisa makes with the
vertical is increasing more and more rapidly.
(e) Upper Midwest firms profit growth slows.
(f) The XYZ Company has been losing money, but will soon
turn this situation around.
36% during 1983.
52. Translate each statement from the following newspaper
column into a statement about derivatives.
(a) In the United States, the ratio R of government debt to na
tional income remained unchanged at around 28? up to
1981, but
(b) then it began to increase more and more sharply, reaching
53. Coffee is poured into the cup shown in Figure 20 at
the rate of 2 cubic inches per secord.The top diameter is 3.5
inches, the bottom diameter is 3 inches, and the height of the
cup is 5 inches. This cup holds about 23 fluid ounces, Determine
the height Ii of the coffee as a function of time t, and sketch
the graph of hQ) from time t = 0 until the time that the cup
is full.