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Unformatted text preview: ues. Moreover,
note that tr H = 4 + 3 + 6 = 13, which we know also equals to the sum of the
three eigenvalues of H . If H has two negative eigenvalues, then the only positive
eigenvalue of H must be greater than 13. Now, it’s easy to check that φ(13) < 0.
This together with the observation that lim +∞ φ( ) = −∞ implies that φ( ) = 0 can
not just have one positive root > 13. Hence H must have no negative eigenvalues.
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Thus the Morse index is 0 and 2 1 1 is a local minimum. 3
E Questions on logical thinking
Consider the function ( )= 2
2 2
2 + 0 2 +
+ 2 2 =0
=0 In the last tutorial we showed that the restriction of this function to a line parallel to
any coordinate axis, is continuous on the entire real line, while on the other hand the
unrestricted function (
) is not continuous at (0 0) (as it has no limit at this point).
(a) Find the partial derivatives
(b) Is the second derivative
S and . (0 0) deﬁned? (a) It’s easy to see that
2 ( 2 − 2)
( 2 + 2 )2
2 ( 2 − 2)
=
( 2 + 2 )2 = Note that ( 0) = (0 2 2 +
+ 2 2 =0
=0 ) = 0 by deﬁnition, so it follows that
4 (0 0) = (0...
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This document was uploaded on 02/10/2014.
 Spring '13

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