Problem Set 3
MH2100
Solutions
1
Questions on understanding the lecture
E
Without doing any calculation, explain why any critical point of the function
(
) = sin
·
arctan(

) + cos(
2

2
)
is degenerate.
S
Since the function is independent of
, all its derivatives with respect to
vanish
and therefore the Hessian matrix looks like
H
=
0
0
0
0
0
It follows that
H
has at least one eigenvalue being
0
so that any critical point of the
function
(
)
is degenerate.
:)
2
Questions on calculation
E
Find critical points and determine their type (degenerate/nondegenerate,
Morse index if there is any) for the following functions:
(a)
(
) = (

+ 1)
2
.
(b)
(
) =
+
50
+
20
, where
>
0
.
(c)
(
) =
ln(
2
+
2
)
.
(d)
(
) =
+
2
4
+
2
+
2
, where
>
0
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
S
(a)
(
) = (

+ 1)
2
. For critical points, we have
= 2(

+ 1) = 0
=

2(

+ 1) = 0
so both equations are

+ 1 = 0
, which is a whole straight line consisting of
critical points.
Since they are not isolated, they must be degenerate.
Since the
value of the function at this line is
0
and
(
)
≥
0
, we see that this line consists of
degenerate minima.
(b)
(
) =
+
50
+
20
, where
>
0
. For critical points, we have
=

50
2
= 0
=

20
2
= 0
so
=
50
2
and
=
20
2
=
20
(50
/
2
)
2
=
4
125
. Now we get
3
= 125
, that is,
= 5
and
= 2
.
Further,
H
=
100
3
1
1
40
3
H
(5 2) =
4
5
1
1
5
and
det
H
= 3
>
0
,
tr
H >
0
. Hence
(5 2)
is a local minimum.
(c)
(
) =
ln(
2
+
2
)
. For critical points, we have
=
ln(
2
+
2
) +
2
2
+
2
= 0
=
ln(
2
+
2
) +
2
2
+
2
= 0
From
= 0
, we see that either
= 0
or
ln(
2
+
2
)+
2
2
2
+
2
= 0
. Let’s consider these
two cases separately.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '13
 Critical Point, hessian matrix, Morse theory

Click to edit the document details