20e-lecture9

20e-lecture9 - Lecture 9 Monday April 20th [email protected]

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Unformatted text preview: Lecture 9 : Monday April 20th [email protected] Key concepts : Local extrema, Hessian, positive definite, determinant Know how to find local extrema using first and second derivatives 9.1 Local extremes For a function f : R n → R defined on an open set U , a point x ∈ U is called a local minimum (respectively, maximum) of f if there exists an open ball B containing x such that f ( y ) ≥ f ( x ) (respectively, f ( y ) ≤ f ( x ) for all y ∈ B . A local extremum of f is either a local minimum or maximum of f . The point x is a critical point of f if ∇ f ( x ) = 0 or if f is not differentiable at x . As in the case of single variable functions, every local extremum of f is a critical point. Proposition. If U ⊂ R n is an open set, and f : U → R n is differentiable, then if x ∈ U is a local extremum of f , ∇ f ( x ) = 0. 9.2 Hessian Matrix In the case of functions of one variable, we test whether a critical point x is a local minimum or maximum or a saddle point of a function...
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20e-lecture9 - Lecture 9 Monday April 20th [email protected]

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