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 ).

Question

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  ).

cdtortosadn · Accepted Answer

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  )  &amp;lt;  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,

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).

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