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Unformatted text preview: 3.133 If is convex ( x 1 + (1 ) x 2 ) ( x 1 ) + (1 ) ( x 2 ) Since is increasing ( ( x 1 + (1 ) x 2 ) ) ( ( x 1 ) + (1 ) ( x 2 ) ) ( ( x 1 ) ) + (1 ) ( ( x 2 ) ) since is also convex. The concave case is proved similarly. 3.134 Let = log . If is convex, ( x ) = ( x ) is an increasing convex function of a convex function and is therefore convex (Exercise 3.133). 3.135 If is positive and concave, then log is concave (Exercise 3.51). Therefore log 1 = log 1 log = log is convex. By the previous exercise (Exercise 3.134), this implies that 1 / is convex. If is negative and convex, then is positive and concave, 1 / is convex, and therefore 1 / is concave....
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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.
- Fall '10