Let f be a continuously differentiable function on [ a , b ] with f ( a ) = f ( b ) and f ′ ( a ) = f ′ ( b ). Then, do there exist x 1 , x 2 ∈ ( a , b ) such that f ′ ( x 1 ) = f ′ ( x 2 ) for x 1 ≠ x 2 ?I think the answer is yes by intermediate value theorem, but am not getting the sufficient rigour to prove it. Any hints. Thanks beforehand.

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Let   f be a continuously differentiable function on   [  a  ,  b  ] with   f  (  a  )  =  f  (  b  ) and       f    ′    (  a  )  =      f    ′    (  b  ). Then, do there exist       x    1    ,      x    2    ∈  (  a  ,  b  ) such that       f    ′    (      x    1    )  =      f    ′    (      x    2    ) for       x    1    ≠      x    2  ?I think the answer is yes by intermediate value theorem, but am not getting the sufficient rigour to prove it. Any hints. Thanks beforehand.

rioolpijpgp · Accepted Answer

By the mean value theorem, there exists   c  ∈  (  a  ,  b  ) s.t.:  0  =            f      (      b      )      −      f      (      a      )              b      −      a        =      f    ′    (  c  )  .Knowing that       f    ′   is continuous, by IVT, there exists       x    1    ∈  (  a  ,  c  ) such that       f    ′    (      x    1    )  =                              f          ′                (        a        )            2      . By the same reasonning, there exists       x    2    ∈  (  c  ,  b  ) such that       f    ′    (      x    2    )  =                              f          ′                (        b        )            2      .Since       f    ′    (  a  )  =      f    ′    (  b  ),       f    ′    (      x    1    )  =      f    ′    (      x    2    ) and       x    1    ≠      x    2   since they belong to disjoints intervals.Note that we could have taken any value between       f    ′    (  a  ) and 0. Choosing                               f          ′                (        a        )            2       and                               f          ′                (        b        )            2       was just for convenience.

Let f be a continuously differentiable function on [ a , b ] with f (

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