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Unformatted text preview: 1 Homework #6 1. Consider f ( x ) = e x for x R . (a) Show that f is convex and compute the Legendre transform f * ( ). Solution It is clear that f 00 ( x ) = e x > 0, hence f is convex. Set = f ( x ) = e x and solve for x : ln = x . Then f * ( ) = x- f ( x ) = ln - f (ln ) = (ln - 1). (b) Prove that e n x e x +( n- 1) e n for all x R , and all positive integers n . Solution Consider the function g ( y ) = e y- y . Then g ( y ) = e y- 1 and has only one root when y = 0. So g 00 ( y ) = e y and g 00 (0) > 0. Hence, y = 0 is global minimum of g , by the second derivative test. Therefore, g ( y ) g (0) = 1 = e y- y 1. Let y = x- n . Then e x- n- x + n 1. Since e n > 0, multiply expression by e n : e x- xe n + ne n e n = xe n e x + ( n- 1) e n , as desired. Also note that Youngs Inequality x f ( x ) + f * ( ) with = e n proves the desired result as well. 2. Consider f ( x,y ) = e ( x 2 + y 2 ) / 2 for ( x,y ) , where = ( x,y ) R 2 | | x | < 1 and | y | < 1 . (a) Show that the Hessian matrix Hf ( x,y ) = f xx f xy f yx f yy is a positive definite matrix for all ( x,y ) ....
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This note was uploaded on 09/09/2009 for the course MATH 410 taught by Professor Staff during the Spring '08 term at Maryland.
- Spring '08