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Unformatted text preview: , ∞ ) and ﬁnd lim p → 0+ f ( p ). Hint: Compute f ( p ) using that f ( p ) = e 1 p ln ( P n k =1  x k  p ) (1) 5. [20 points] Let x = ( x 1 , ··· ,x n ) ∈ R n and g ( p ) = 1 n n X k =1  x  p k ! 1 /p . (1) Using H¨ older’s inequality show that g ( p ) is increasing on (0 , ∞ ). (2) Find lim p →∞ g ( p ). (3) Suppose that x k 6 = 0 for all k . Show that lim p → 0+ g ( p ) = (  x 1  x 2 ··· x n  ) 1 /n , (4) Show that (1) and (3) imply the inequality of geometric and arithmetic means, that is, (  x 1  x 2 ··· x n  ) 1 /n ≤  x 1  +  x 2  + ··· +  x n  n . 2...
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 Spring '09
 Integers, Order theory, Monotonic function, 1 j, 0 1 k

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