Macroeconomics Exam Review 208

Macroeconomics Exam Review 208 - 4.34 Define ( ) = ( ) ( )...

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Unformatted text preview: 4.34 Define ( ) = ( ) ( ) ( ) ( ) Then is continuous on [ , ] and differentiable on ( , ) with ( ) = ( ) ( ) ( ) ( ) ( ) = ( ) By Rolles theorem (Exercise 5.8), there exists ( , ) such that ( ) = ( ) ( ) ( ) = 0 4.35 Assume ( x ) for every x . By the mean value theorem, for any x 2 x 1 in , there exists x ( x 1 , x 2 ) such that ( x 2 ) = ( x 1 ) + [ x ]( x 2 x 1 ) Using (4.6) ( x 2 ) = ( x 1 ) + =1 ( x )( 2 1 ) (4.44) ( x ) and x 2 x 1 implies that =1 ( x )( 2 1 ) and therefore ( x 2 ) (...
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