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Unformatted text preview: f ( x 1 ) ∥ x 3 − x 1 ∥ ≥ f ( x 2 ) − f ( x 1 ) ∥ x 2 − x 1 ∥ . 3. Let C ⊂ R n be a convex set and let f ( x ) = h ( g ( x )), where g : C → R and h : R → R . (a) Let f ( x ) = e g ( x ) . If g ( x ) is convex, then f ( x ) is convex. True or false? Why? (b) Let f ( x ) = 1 g ( x ) . If g is convex and positive, then f is convex. True or false? Why? 1 4. (a) Show that x 4 + 3 y 4 ≤ √ ln ( e x 2 4 + 3 4 e y 2 ) for all positive number x and y . (Hint: The desired inequality follows from the convexity of an appropriate function.) (b) Show that ± x 2 + y 3 + z 12 + w 12 ² 4 ≤ 1 2 x 4 + 1 3 y 4 + 1 12 z 4 + 1 12 w 4 with equality if and only if x = y = z = w . 2...
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This document was uploaded on 09/11/2011.
 Spring '09

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