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Unformatted text preview: Math 3110 Spring 09 Homework 9: Selected Solutions [12-3]: Claim : Let f : [- a,a ] R be continuous, with f (0) > f (- a ) ,f ( a ). In this case there exist x 1 ,x 2 [- a,a ] such that a horizontal chord of length exactly a joins ( x 1 ,f ( x 1 )) to ( x 2 ,f ( x 2 )). Proof Let g : [- a, 0] R be defined by g ( x ) = f ( x + a )- f ( x ). The function g ( x ) is clearly continuous on its domain and g ( x ) = 0 exactly when a horizontal chord of length a joins f ( x ) to f ( x + a ). Since g (- a ) = f (0)- f (- a ) > 0 and g (0) = f ( a )- f (0) < 0, Bolzanos Theorem tells us that g ( x ) = 0 for some x [- a, 0], or equivalently that there is a horizontal chord of length exactly a joining two points on the graph of f ([- a,a ]). [12.5]: Claim : An equilateral triangle can be inscribed in any smooth convex closed curve C in R 2 . Proof [See Figure 2 on last page.] Choose any point P on C . Because C is both smooth and convex, there is a tangent line T P to C at P whose only intersection with C occurs at P . Let L 1 , L 2 : [0 , 1] ( , 2 3 ) R 2 describe line segments originating at P , making angles and ( + 3 ) respectively with T P and ending at their points of first intersection Q ( ) = L 1 (1 , ) and R (...
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