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Unformatted text preview: , 3] R given by f ( x ) = x ( x2)( x4) . 8. Let a 1 , a 2 , ..., a n be real numbers and let f be dened on R by f ( x ) := n X i =1 ( a ix ) 2 for x R . Find the unique point of relative minimum for f . 9. Let f, g be dierentiable on R and suppose that f (0) = g (0) and that f ( x ) 6 g ( x ) for all x > 0. Show that f ( x ) 6 g ( x ) for all x > 0. 10. Prove that the polynomial f m ( x ) = x 33 x + m never has two roots in [0 , 1], no matter what m is. 2 11. Let f be continuous on [ a,b ] and dierentiable on ( a,b ). Suppose that f 2 ( b )f 2 ( a ) = b 2a 2 . Prove (using Rolles theorem) that ( x ( a,b ) )( f ( x ) f ( x ) = x ) . 12. Using the Mean Value Theorem prove that ( (1 , 0) ) 1 + 2 1 + < 1 + < 1 + 2...
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 Winter '10
 Spyros

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