hw3a 475 - ± ▀ 1 Note that these proofs can be found in...

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1 ECON475 HW #3 Answers Proof 1 We will show that if g G is not the global maximum of ± , then ± must have another critical point on ² . Suppose there is a point ³ G in ² with ±´³ G µ ¶ ±´g G µ . Suppose further that (w.l.o.g.), ³ G ¶ g G . Since ± is decreasing just to the right of g G and eventually increasing again somewhere between g G and ³ G , say · G . But then · G is an interior local minimum of ± , and therefore is a critical point other than g G – contradicting the hypothesis that g G is the only critical point of ± . Therefore, g G is the global maximum of ± on its domain ² . Proof Suppose ± ¸¸ is always positive (negative) on domain ² . By Theorem 3.2, ± ¸ is an increasing (decreasing) function on ² . This means that ± ¸ can be zero at at most one point. If there is a point g G where ± ¸ ´g G µ ¹ 0 , then g G is a local minimum (maximum) of ± since ± ¸¸ ´g G µ ¶ 0 ( ± ¸¸ ´g G µ º 0µ . By Theorem 3.5, g G is the global minimum (maximum) of
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Unformatted text preview: ± . ▀ 1 Note that these proofs can be found in Chapter 3 of Simon & Blume (1994). Theorem 3.5 Suppose that: (a) the domain of ± is an interval ² (finite or infinite) in » ¼ (b) g G is a local maximum of ± , and (c) g G is the only critical point of ± on ² . Then, g G is the global maximum of f on ² . Theorem 3.5 If ± is a ½ ¾ function whose domain is an interval ² and if ± ¸¸ is never zero on ² , then ± has at most one critical point in ² . This critical point is a global minimum if ± ¸¸ ¶ 0 , and a global maximum if ± ¸¸ º 0 . Theorem 3.2 Let ± be a ½ ¼ function on domain ¿ À Á . If ± ¸ ¶ 0 ( ± ¸ º 0µ on interval ´Â, õ À ¿ , then ± is increasing (decreasing) on ´Â, õ. If ± is increasing (decreasing) on ´Â, õ , then ± ¸ Ä 0 ( ± ¸ Å 0µ on ´Â, õ . A good answer to question 1: 2...
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This note was uploaded on 05/26/2010 for the course ECON 475 taught by Professor Voyvoda during the Spring '10 term at Middle East Technical University.

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hw3a 475 - ± ▀ 1 Note that these proofs can be found in...

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