Let f:[0,2]->R be a continuous function such that f(0)=f(2). Use the intermediate value theorem to prove that there exist numbers x,y in[0,2] such that f(x)=f(y) and |x−y|=1. Hint: Introduce the auxiliary function g:[0,1]->R defined by g(x)=f(x+1)−f(x).

Paola Mayer 2022-10-22 Answered
Let f : [ 0 , 2 ] R be a continuous function such that f ( 0 ) = f ( 2 ). Use the intermediate value theorem to prove that there exist numbers x , y [ 0 , 2 ] such that f ( x ) = f ( y ) and | x y | = 1.
Hint: Introduce the auxiliary function g : [ 0 , 1 ] R defined by g ( x ) = f ( x + 1 ) f ( x ).
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Answers (1)

cdtortosadn
Answered 2022-10-23 Author has 19 answers
The given function g is continous in [ 0 , 1 ] and
{ g ( 0 ) = f ( 1 ) f ( 0 ) g ( 1 ) = f ( 2 ) f ( 1 ) g ( 0 ) g ( 1 ) < 0 , unless f ( 1 ) = f ( 0 ) (why?)
and so by the IVM there exists c ( 0 , 1 ) s.t. g ( c ) = 0 end the exercise now,
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