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Unformatted text preview: 363 3.8. LOCAL EXTREMA
3. Q(x, y ) = x2 − y 2 has
A = [Q] = 10
0 −1 so
∆2 = − 1 < 0
and Q is not deﬁnite.
4. Q(x, y ) = xy has
A = [Q] = 01
10 so
∆2 = − 1 < 0
and Q is not deﬁnite.
5. Finally, for the one we couldn’t decide in an obvious way:
Q(x, y ) = 2x2 − 2xy + 3y 2 has
A = [Q] = 2 −1
−1 3 so
∆1 = 2 > 0
∆2 = 5 > 0
and Q is positive deﬁnite.
When applied to the Hessian of f : R2 → R, the matrix representative of the
Hessian form is the matrix of partials of f , sometimes called the Hessian
matrix of f :
→
→
fxx (− ) fxy (− )
a
a
→
Hf (− ) =
a
→
− ) f (− ) .
→
fxy ( a
yy a
this gives us 17 17
The Second Derivative Test was published by Joseph Louis Lagrange (17361813) in
his very ﬁrst mathematical paper [33] ([20, p. 323]). ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

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