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Unformatted text preview: -1 / 2. By the Schwarz inequality, we therefore conclude 1 4 = Z b a xf ( x ) f ( x )dx Z b a | f ( x ) | 2 ! 1 / 2 Z b a | xf ( x ) | 2 dx ! 1 / 2 . (1) The equality holds in this inequality if and only if the integrands are proportional. That means f ( x ) = cxf ( x ) for some c 6 = 0, since f is not identically zero. All solutions to this dierential equation have the form f ( x ) = de cx 2 / 2 , 1 where d is another nonzero constant. For such a function though we cannot have f (0) = f (1) = 0, so equality in (1) is impossible. Thus Z b a ( f ( x )) 2 dx Z b a x 2 f 2 ( x )dx > 1 / 4 . 2...
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- Spring '09