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**Unformatted text preview: **Final Exam Questions Econ 600, Summer 2011 Leland Crane and Jonathan Kreamer Read the entire exam before starting. Show your work and clearly mark your nal answer for each question. Good luck! 1. (10 points) Let f : R → R and g : R → R . Let h : R → R be de ned as h ( x ) = min ( f ( x ) ,g ( x )) . (a) If f and g are concave, is h concave? Prove or o er a counter-example. (b) If f and g are quasiconcave, is h quasiconcave? Prove or o er a counter-example. 2. (5 points) Suppose that f : R → R is continuous and invertible. Prove that f is either strictly increasing or strictly decreasing. 3. (5 points) Suppose f : A ⊂ R n → R is strictly quasi-concave and has a local minimum at x ? ∈ A . Prove A is not open. 4. (15 points) Let U ( y ) = max x f ( x ) s.t. x ≤ G ( y ) where f and G are concave. Prove that U is concave. 5. (15 points) Let f : R → R . Suppose you are given that on the interval [ x 1 ,x 2] , the absolute value of the derivative of f never exceeds some positive constant...

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- Fall '09
- Economics, Monotonic function, Convex function, Leland Crane