Engineering Calculus Notes 375

Engineering Calculus Notes 375 - 363 3.8. LOCAL EXTREMA 3....

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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 (1736-1813) 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.

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