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bmhicks_homework_4

# bmhicks_homework_4 - Brandy Hicks MTHSC 974.001 Homework 4...

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Brandy Hicks September 19, 2007 MTHSC 974.001 - Homework # 4 1. Let f and g be convex functions in the modern sense. Let x be an interior point of the domain of f + g . Prove: (a) D - ( f + g )( x ) D - f ( x ) + D - g ( x ) Proof: Let f,g conV ( R ). Then let x 0 int(dom( f + g )), so x 0 int(dom( f )) and x 0 int(dom( g )), Hence dom( f + g )( x 0 ) = dom f ( x 0 ) dom g ( x 0 ) , Now as we proved in class, if x 0 int(dom( f )), then D - f ( x 0 ) = lim x x 0 f ( x ) - f ( x 0 ) x - x 0 = sup x<x 0 f ( x ) - f ( x 0 ) x - x 0 < similarly D + f ( x 0 ) = lim x x 0 f ( x ) - f ( x 0 ) x - x 0 = inf x>x 0 f ( x ) - f ( x 0 ) x - x 0 > -∞ Then, since x 0 int(dom( f + g )) and dom( f + g ) = dom f dom g D - ( f + g )( x 0 ) = lim x x 0 ( f + g )( x - x 0 ) x - x 0 = sup x<x 0 ( f + g )( x - x 0 ) x - x 0 Clearly since sup x<x 0 ( f + g )( x ) sup x<x 0 f ( x ) + sup x<x 0 g ( x ) then D - f ( x 0 ) + D - g ( x 0 ) D - ( f + g )( x 0 ) sup x<x 0 ( f + g )( x ) - ( f + g )( x 0 ) x - x 0 = D - ( f + g )( x 0 ) D - ( f + g )( x ) D - f ( x ) + D - g ( x ) 1

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(b) D + ( f + g )( x ) D + f ( x ) + D + g ( x ) Proof: Let f,g conV ( R ). Then let x 0 int(dom( f + g )), so x 0 int(dom( f )) and x 0 int(dom(
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bmhicks_homework_4 - Brandy Hicks MTHSC 974.001 Homework 4...

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