{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Macroeconomics Exam Review 212

# Macroeconomics Exam Review 212 - c 2001 Michael Carter All...

This preview shows page 1. Sign up to view the full content.

4.45 Assume not. That is, assume that y = 𝑓 ( x 1 ) 𝑓 ( x 2 ) ∕∈ conv 𝐴 Then by the (strong) separating hyperplane theorem (Proposition 3.14) there exists a linear functional 𝜑 on 𝑌 such that 𝜑 ( y ) > 𝜑 ( a ) for every a 𝐴 (4.48) where 𝜑 ( 𝑦 ) = 𝜑 ( 𝑓 ( x 1 ) 𝑓 ( x 2 )) = 𝜑 ( 𝑓 ( x 1 )) 𝜑 ( 𝑓 ( x 2 )) 𝜑𝑓 is a functional on 𝑆 . By the mean value theorem (Theorem 4.1), there exists some ¯ x [ x 1 , x 2 ] such that 𝜑 𝑓 ( x 1 ) 𝜑 𝑓 ( x 2 )) = 𝐷 ( 𝜑 𝑓 )[¯ x ]( x 1 x 2 ) = 𝜑 𝐷𝑓 x ]( x x 2 ) = 𝜑 ( 𝑎 ) for some 𝑎 𝐴 contradicting (4.44). 4.46 Define : [ 𝑎, 𝑏 ] → ℜ by ( 𝑥 ) = ( 𝑓 ( 𝑏 ) 𝑓 ( 𝑎 ) ) 𝑔 ( 𝑥 ) ( 𝑔 ( 𝑏 ) 𝑔 ( 𝑎 ) ) 𝑓 ( 𝑥 ) 𝐶 [ 𝑎, 𝑏 ] and is differentiable on 𝑎, 𝑏 ) with ( 𝑎 ) = ( 𝑓 ( 𝑏 ) 𝑓 ( 𝑎 ) ) 𝑔 ( 𝑎 ) ( 𝑔 ( 𝑏 ) 𝑔 ( 𝑎 ) ) 𝑓 ( 𝑎 ) = 𝑓 ( 𝑏 ) 𝑔 ( 𝑎 ) 𝑓 ( 𝑎 ) 𝑔 ( 𝑏 ) = ( 𝑏 ) By Rolle’s theorem (Exercise 5.8), there exists 𝑥 ( 𝑎, 𝑏 ) such that ( 𝑥 ) = ( 𝑓 ( 𝑏 ) 𝑓 ( 𝑎
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern