hw9 - Math508 Fall 2010 Jerry L Kazdan Problem Set 9 D UE...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 9 DUE: Thurs. Nov. 18, 2010. Late papers will be accepted until 1:00 PM Friday . Note: We say a function is smooth if its derivatives of all orders exist and are continuous. 1. Let f ( x ) be a smooth function for x 1 with the property that f 0 ( x ) 0 as x . a) Show that f ( n + 1 ) - f ( n ) 0 as n . b) Compute lim n ± 5 n + 1 - 5 n ² . 2. Find a continous function f and a constant C so that Z 2 x 0 f ( t ) dt = 2 x cos x + e 4 x + C . 3. Let f : [ 0 , 1 ] R be a continuous function. a) If R 1 0 f ( x ) dx = 0, prove that f ( x ) is positive somewhere and negative somewhere in this interval (unless it is identically zero). b) Use this to show that k f k 1 : = Z 1 0 | f ( x ) | dx is a norm on C ([ 0 , 1 ]) . c) Show that C ([ 0 , 1 ]) with this norm is not complete. 4. Let f ( x ) C ([ a , b ]) . Show that exp ³ 1 b - a Z b a f ( x ) dx ´ 1 b - a Z b a exp [ f ( x )] dx [HINT: Use the inequality e u 1 + u

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hw9 - Math508 Fall 2010 Jerry L Kazdan Problem Set 9 D UE...

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