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Math 508
Exam 2
Jerry L. Kazdan
December 8, 2006
12:00 – 1:20
Directions
This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points),
Part B has 5 traditional problems (15 points each, so 75 points).
Closed book, no calculators – but you may use one 3
00
×
5
00
card with notes.
Part A: Short Problems
(3 problems, 8 points each).
A–1. A continuous function
f
:
R
→
R
has the property that
Z
x
0
f
(
t
)
dt
= cos(
x
)
e

x
+
C,
where
C
is some constant. Find both
f
(
x
) and the constant
C
.
A–2. A function
h
:
R
→
R
with two continuous derivatives has the property that
h
(0) = 2,
h
(1) = 0, and h(3)=1. Prove there is at least one point
c
in the interval 0
< x <
3 where
h
00
(
c
)
>
0 by finding some
explicit
m >
0 (such as
m
= 2
/
3) with
h
00
(
c
)
≥
m
.
A–3. Say a smooth function
u
(
x
) satisfies
u
00

c
(
x
)
u
= 0
for 0
≤
x
≤
1 (here
c
(
x
) is some given
contunuous function).
If
c
(
x
)
>
0 everywhere, show that there is
no
point where
u
(
x
) is both positive
and
has a
local maximum.
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 Fall '10
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 Math, Topology, Derivative, Continuous function, Metric space, Traditional problems, Jerry L. Kazdan

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