Macroeconomics Exam Review 208

Macroeconomics Exam Review 208 - Solutions for Foundations...

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4.34 Define ( 𝑥 ) = 𝑓 ( 𝑥 ) 𝑓 ( 𝑏 ) 𝑓 ( 𝑎 ) 𝑏 𝑎 ( 𝑥 𝑎 ) Then is continuous on [ 𝑎, 𝑏 ] and differentiable on ( 𝑎, 𝑏 ) with ( 𝑏 ) = 𝑓 ( 𝑏 ) 𝑓 ( 𝑏 ) 𝑓 ( 𝑎 ) 𝑏 𝑎 ( 𝑏 𝑎 ) 𝑓 ( 𝑎 ) = ( 𝑎 ) By Rolle’s theorem (Exercise 5.8), there exists 𝑥 ( 𝑎, 𝑏 ) such that ( 𝑥 ) = 𝑓 ( 𝑥 ) 𝑓 ( 𝑏 ) 𝑓 ( 𝑎 ) 𝑏 𝑎 = 0 4.35 Assume 𝑓 ( x ) 0 for every x 𝑋 . By the mean value theorem, for any x 2 x 1 in 𝑋 , there exists ¯ x ( x 1 , x 2 ) such that 𝑓 ( x 2 ) = 𝑓 ( x 1 ) + 𝐷𝑓 x ]( x 2 x 1 ) Using (4.6) 𝑓 ( x 2 ) = 𝑓 ( x 1 ) + 𝑛 𝑖 =1 𝐷 𝑥 𝑖 𝑓 x )( 𝑥 2 𝑖 𝑥 1 𝑖 ) (4.44) 𝑓 x ) 0 and x 2 x 1 implies that 𝑛 𝑖 =1 𝐷 𝑥 𝑖 𝑓 x )( 𝑥 2 𝑖 𝑥 1 𝑖 ) 0 and therefore 𝑓 ( x 2 ) 𝑓 ( x 1 ). 𝑓 is increasing. The converse was established in Exercise 4.15 4.36 𝑓 x ) > 0 and x 2 x 1 implies that 𝑛 𝑖 =1 𝐷 𝑥 𝑖 𝑓 x )( 𝑥 2 𝑖 𝑥 1 𝑖 ) > 0 Substituting in (4.44) 𝑓 ( x 2 ) = 𝑓 ( x 1 ) + 𝑛 𝑖 =1 𝐷 𝑥 𝑖 𝑓 x )( 𝑥 2 𝑖 𝑥 1 𝑖 ) > 𝑓 ( x 1 ) 𝑓 is strictly increasing. 4.37 Differentiability implies the existence of the gradient and hence the partial deriv- atives of
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