18.06 Problem Set 8
Due Wednesday, 23 April 2008 at 4 pm in 2106.
Problem 1:
Do problem 3 in section 6.5 (pg. 339) in the book.
Problem 2:
Do problem 6 in section 6.5 (pg. 339).
Problem 3:
For what numbers
c
and
d
are the matrices
A
and
B
positive definite?
Test the 3 determinants:
A
=
c
2
3
2
c
4
3
4
1
B
=
1
2
1
2
d
3
1
3
1
Problem 4:
Do problem 15 in section 6.5 (pg. 340).
Problem 5:
Do problem 28 in section 6.5 (pg. 342).
Problem 6:
a) Let
f
1
(
x, y
) =
1
4
x
4
+
x
2
y
+
y
2
. Find the second derivative matrix
A
1
=
∂
2
f/∂x
2
∂
2
f/∂x∂y
∂
2
f/∂y∂x
∂
2
f/∂y
2
A
1
is not positive definite everywhere  find the conditions on
x
and
y
for it to
be positive definite. (Interesting question: check what happens when both partial
derivatives vanish, that is, when
∂f/∂x
=
∂f/∂y
= 0. Is
f
still positive definite? It
turns out that
f
1
does attain a minimal value, but not along isolated points. Find
the points where it hits a global minimum. This part is not required.)
b) Let
f
2
(
x, y
) =
x
3
+
xy

x
. Find the second derivative matrix
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 Spring '08
 CHANILLO
 Derivative, Matrices, potential solution, positive definite

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