MATH 242 Analysis 1 (Fall, 2011)
Assignment 4. Due in class Tuesday, November 8.
Instructions: This assignment is out of 50 points. You may choose any questions
whose points add up to 50. As in the previous assignment, provide all details in your
work. Also as previously, stars (*) indicate possibly more challenging questions.
1.
(10 pts.)
(a) Let
f
(
x
) = (
x
−
2)(
x
−
3)(
x
−
4)(
x
−
5)(
x
−
6). If the Bisection Method
is applied, beginning with the interval [0
,
7], which of the roots of
f
is
located ?
(b) Show that the equation
1
√
x
2
+
x
+
x
2
= 2
x
has at least two solutions
x >
0.
2.
(10 pts.)
Let
f
: [0
,
1]
→
R
be continuous and such that
f
(0) =
f
(1).
Prove that there exists a point
c
∈
[0
,
1
2
] such that
f
(
c
) =
f
(
c
+
1
2
). [Hint:
Consider
g
(
x
) =
f
(
x
)
−
f
(
x
+
1
2
).] Conclude that there are, at any time,
antipodal points on the earth’s equator that have the same temperature.
3.
(10 pts.)
Let
a < b
and
f
: (
a,b
)
→
R
be continuous and such that
lim
x
→
a
f
(
x
) = 0 = lim
x
→
b
f
(
x
).
Prove that
f
is bounded on (
a,b
) and
attains either a maximum or a minimum on (
a,b
).
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 Spring '08
 Drury
 Math, Calculus, Continuous function, 10 pts, Uniform continuity, 20 pts, antipodal

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