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Unformatted text preview: f : R 3 R has a local maximum (or local minimum) at a , then d a f ( v ) = 0 for all v R 3 . The converse is not true: for example, the function f ( x ) = x 3 has a critical point at the origin but is strictly increasing on the whole real line. Other phenomena are possible for multivariate functions: for example, the restriction of f ( xy ) = x 2 y 2 to the xaxis has a minimum at the origin, while its restriction to the yaxis has a maximum there. So to determine whether a critical point a is a local extremum, we need to study the (local) behavior in all directionsin particular, we need to study the Hessian d 2 a f ....
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 Fall '08
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 Calculus, Derivative

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