Brandy Hicks
August 31, 2007
MTHSC 974.001  Homework # 2
1. For
r > o
, draw the graph of
f
1
r
(
X
) =
1
2
rx
2
,

x
 ≤
1
r

x
 
1
2
r
,

x
 ≥
1
r
to see if
f
1
r
is convex on
R
.
1.JPG
Although the drawing is rudimentary, it appears that
f
1
r
(
x
) is convex on
R
(a) What happens as
r
→
+
∞
?
Proof:
As
r
→ ∞
,
f
1
r
(
x
)
→ ∞
. As
r
gets very large,
1
r
gets very close to zero.
Since the value of
f
1
r
(
x
) =
1
2
rx
2
when

x
 ≤
1
r
, then when the absolute value of
x
is
between 0 and
1
r
, which is very close to 0,
we get
1
2
·
r
(which remember is approaching
∞
)
·
x
2
(and
x
is very close to 0).
This means that the equation basically becomes
1
2
· ∞ ·
1
×
10

100000000000
.
Therefore this whole piece of the function is going to 0.
Now we look at the the other segment of the function. We see that when

x
 ≥
1
r
,

x
 
1
2
·
r
.
So

x

has to be greater than
1
r
which is basically
1
∞
, which is very close to zero.
This then makes our function

x

(which is greater than 0)

1
2
·∞
.
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 Fall '07
 Peterson
 Convex set, Empty set, Long and short scales, Convex function

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