Macroeconomics Exam Review 208

Macroeconomics Exam Review 208 - 4.34 Define ℎ = − −...

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Unformatted text preview: 4.34 Define ℎ ( ) = ( ) − ( ) − ( ) − ( − ) Then ℎ is continuous on [ , ] and differentiable on ( , ) with ℎ ( ) = ( ) − ( ) − ( ) − ( − ) ( ) = ℎ ( ) By Rolle’s theorem (Exercise 5.8), there exists ∈ ( , ) such that ℎ ′ ( ) = ′ ( ) − ( ) − ( ) − = 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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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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