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410HW7

# 410HW7 - 1 Homework#6 1 Consider f(x = ex for x R(a Show...

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1 Homework #6 1. Consider f ( x ) = e x for x R . (a) Show that f is convex and compute the Legendre transform f * ( ξ ). Solution It is clear that f ( x ) = e x > 0, hence f is convex. Set ξ = f ( x ) = e x and solve for x : ln ξ = x . Then f * ( ξ ) = - f ( x ) = ξ ln ξ - f (ln ξ ) = ξ (ln ξ - 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 ( y ) = e y and g (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 Young’s Inequality f ( x ) + f * ( ξ ) with ξ = e n proves the desired result as well. 2. Consider f ( x, y ) = e ( x 2 + y 2 ) / 2 for ( x, y ) Ω, where Ω = ( 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 ) Ω. Solution One way to show the matrix is positive definite is to show all the principle minors are positive. In this case, show that f xx > 0 and f xx f yy - f 2 xy > 0. Then f xx = e x

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410HW7 - 1 Homework#6 1 Consider f(x = ex for x R(a Show...

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