practicefinal1 - f ( c ) = c. (4) Let f : X R be a...

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(1) Let f : [ a,b ] R such that f 0 exists on [ a,b ] and f 00 exists on ( a,b ) . Assume f ( a ) < 0 and f ( b ) > 0 . Assume f 00 ( x ) 0 for all x. Prove there exists a unique c ( a,b ) such that f ( c ) = 0 . (2) Describe all continuous functions f : R R such that the image of f is contained in {- 1 , 1 } . (3) Let f : X X be a continuous map on a complete, non-empty metric space such that d ( f ( x ) ,f ( y )) < 1 2 d ( x,y ) for all points x,y X. Prove there exists c X such that
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Unformatted text preview: f ( c ) = c. (4) Let f : X R be a uniformly continuous function and X a metric space. Prove that if ( a n ) is a Cauchy sequence in X, then ( f ( a n )) converges in R . (5) Let f : [ a,b ] R be a continuous function. Prove there exists c ( a,b ) such that f ( c ) = 1 b-a R b a f ( t ) dt. 1...
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