Let a , b ∈ R , a &lt; b and let f be a differentiable real-valued function on an open subset of R that contains [a,b]. Show that if γ is any real number between f ′ ( a ) and f ′ ( b ) then there exists a number c ∈ ( a , b ) such that γ = f ′ ( c ).Hint: Combine mean value theorem with the intermediate value theorem for the function ( f ( x 1 ) − f ( x 2 ) ) x 1 − x 2 on the set { ( x 1 , x 2 ) ∈ E 2 : a ≤ x 1 &lt; x 2 ≤ b }.I am having a lot of trouble trying to start on this problem.

Question

Let   a  ,  b  ∈      R  ,   a  &amp;lt;  b and let   f be a differentiable real-valued function on an open subset of       R   that contains [a,b]. Show that if   γ is any real number between       f    ′    (  a  ) and       f    ′    (  b  ) then there exists a number   c  ∈  (  a  ,  b  ) such that   γ  =      f    ′    (  c  ).Hint: Combine mean value theorem with the intermediate value theorem for the function             (      f      (              x        1            )      −      f      (              x        2            )      )                      x        1            −              x        2             on the set   {  (      x    1    ,      x    2    )  ∈      E    2    :  a  ≤      x    1    &amp;lt;      x    2    ≤  b  }.I am having a lot of trouble trying to start on this problem.

Lankenp19hh · Accepted Answer

Let  g  (      x    1    ,      x    2    )  =            f      (              x        1            )      −      f      (              x        2            )                      x        1            −              x        2            Then      lim                  x        2            ↘      a        g  (  a  ,      x    2    )  =      f    ′    (  a  )   and       lim                  x        1            ↗      b        g  (      x    1    ,  b  )  =      f    ′    (  b  )So if   ε is sufficiently small, what can you say about   g  (  a  ,  a  +  ε  )  ,  γ and   g  (  b  −  ε  ,  b  )?Now if you define   h  (  t  )  =  g  (  a  +  (  b  −  ϵ  −  a  )  t  ,  a  +  ϵ  +  (  b  −  a  −  ϵ  )  t  ), what does the Intermediate Value Theorem tell you?

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