lect12_7

lect12_7 - Extrema of Functions of Several Variables -...

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Extrema of Functions of Several Variables - (12.7) 1. Local Extremes and Critical Points Definition: Definition: The point ! a , b " is a critical point of a function f ! x , y " if ! a , b " is in the domain of f and either ! f ! a , b " ! 0 " or ! f ! a , b " DNE ! one or both of f x ! a , b " and f y ! a , b " don&t exist) A necessary condition of a local extremum: If f ! x , y " has a local extremum at ! a , b " , then ! a , b " must be a critical point of f . Warning!! A critical point of f may not be a local extremum. Saddle points: The point a , b , f ! a , b " is a saddle point of z ! f ! x , y " if ! a , b " is a critical point of f and if every open disk centered at ! a , b " in the domain of f contains points ! x , y " for which f ! x , y " # f ! a , b " and points ! x , y " for which f ! x , y " $ f ! a , b " . Example Let f ! x , y " ! xe " x 2 /2 " y 3 /3 % y . Find all critical points of f and determine graphically if they are local extrema. f x ! e " x 2 /2 " y 3 /3 % y % x ! " x " e " x 2 /2 " y 3 /3 % y ! 0 f y ! x ! " y 2 % 1 " e " x 2 /2 " y 3 /3 % y ! 0 # 1 " x 2 ! 0 x ! 0or1 " y 2 ! 0 x !& 1 and x ! 0or y 1 # critical points: !
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lect12_7 - Extrema of Functions of Several Variables -...

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