Let f ( x ) be a continuous function on [ a , b ] and f ′ ( a ) exists. Let ξ be a number such that f ′ ( a ) &gt; ξ &gt; f ( b ) − f ( a ) b − a Prove that there is a c ∈ ( a , b ) such that f ( c ) − f ( a ) c − a = ξI try to use Intermediate Value Theorem to show this. I let g ( x ) = f ( x ) − f ( a ) x − a . I try to show this function is continuous on [ a , b ] but I don know how to show it continuous at endpoint.

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Let   f  (  x  ) be a continuous function on   [  a  ,  b  ] and       f    ′    (  a  ) exists. Let   ξ be a number such that      f    ′    (  a  )  &amp;gt;  ξ  &amp;gt;            f      (      b      )      −      f      (      a      )              b      −      a      Prove that there is a   c  ∈  (  a  ,  b  ) such that            f      (      c      )      −      f      (      a      )              c      −      a        =  ξI try to use Intermediate Value Theorem to show this. I let   g  (  x  )  =            f      (      x      )      −      f      (      a      )              x      −      a      . I try to show this function is continuous on   [  a  ,  b  ] but I don know how to show it continuous at endpoint.

basquinas6v · Accepted Answer

We have that the mapping:  h  (  x  )  :=      {                            g          (          x          )                          :          x          ≠          a                                                  f            ′                    (          a          )                          :          x          =          a                        is continuous on   [  a  ,  b  ]. Thus we can apply the IVT to   h; I presume you can take it from there.

Let f ( x ) be a continuous function on [ a , b ] and f

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