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Unformatted text preview: Honors Introduction to Analysis I Homework IX Solution April 21, 2009 Problem 1 Suppose f is C 1 on an interval and that f satisfies the Holder condition of order ,  f ( x ) f ( y )  M  x y  for all x,y in the interval, where (0 , 1] is fixed. Show that  2 h f ( x )  c  h  1+ . How does the constant c relate to M ? Solution. We apply the mean value theorem to g ( t ) = f ( t + h ) f ( t ) on the interval [ x,x + h ]. Then 2 h f ( x ) = f ( x + 2 h ) 2 f ( x + h ) + f ( x ) = f ( x + 2 h ) f ( x + h ) ( f ( x + h ) f ( x )) = g ( x + h ) g ( x ) and g ( t ) = f ( t + h ) f ( t ). By the MVT we get that g ( x + h ) g ( x ) = hg ( x ) for some x ( x,x + h ), which is equivalent to saying 2 h f ( x ) = h ( f ( x + h ) f ( x )) so  2 h f ( x )  =  h  f ( x + h ) f ( x )   h  M  x + h x  = M  h  1+ c  h  1+ for c M . Hence  2 h f ( x )  c  h  1+ for c M (the minimal value for c , for which this is true, is M ) Problem 2 Suppose that f has a zero of order j at x and that g has a zero of order k at x . What can you say about the order of zero of the function f + g , f g and f/g at x ? Solution. By definition f ( x ) = a j ( x x ) j + o (  x x  j ) and g ( x ) = a k ( x x ) j + o (  x x  k ), or f ( n ) ( x ) = 0 for n = 0 ,...,j 1 and f ( j ) ( x ) 6 = 0 and g ( n ) ( x ) = 0 for n = 0 ,...,k 1 and g ( k ) ( x ) 6 = 0. Let m = min { j,k } , then ( f + g ) ( n ) ( x ) = 0 for n = 0 ,...,m 1 and ( f + g ) ( m ) ( x ) 6 = 0. Or ( f + g )( x ) = a m ( x x ) m + o (  x x  m ), so f + g has a zero of order min { j,k } at x . For any n , ( fg ) ( n ) ( x ) = n i =0 a i f ( i ) ( x ) g ( n i ) ( x ) for a i = n i . From this we notice that ( fg ) ( n ) ( x ) = 0 for n = 0 ,...,k + j 1 and ( fg ) ( k + j ) ( x ) = k + j j f ( j ) ( x ) g ( k ) ( x ) 6 = 0, so fg has a zero of order j + k . Equivalently one can see that ( fg )( x ) = a k a j ( x x ) j + k + o (  x x  k + j ). Lets first analyze the case when j < k ....
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This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell University (Engineering School).
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