Suppose that a function f ( x ) is differentiable ∀ x ∈ [ a , b ]. Prove that f ′ ( x ) takes on every value between f ′ ( a ) and f ′ ( b ).If the above question is a misprint and wants to say "prove that f ( x ) takes on every value between f ( a ) and f ( b )", then I have no problem using the intermediate value theorem here.If, on the other hand, it is not a misprint, then it seems to me that I can't use the Intermediate value theorem, as I can't see how I am authorised to assume that f ′ ( x ) is continuous on [ a , b ].Or perhaps there is another way to look at the problem?

Question

Suppose that a function   f  (  x  ) is differentiable   ∀  x  ∈  [  a  ,  b  ]. Prove that       f    ′    (  x  ) takes on every value between       f    ′    (  a  ) and       f    ′    (  b  ).If the above question is a misprint and wants to say &quot;prove that   f  (  x  ) takes on every value between   f  (  a  ) and   f  (  b  )&quot;, then I have no problem using the intermediate value theorem here.If, on the other hand, it is not a misprint, then it seems to me that I can&#039;t use the Intermediate value theorem, as I can&#039;t see how I am authorised to assume that       f    ′    (  x  ) is continuous on   [  a  ,  b  ].Or perhaps there is another way to look at the problem?

Arcatuert3u · Accepted Answer

This is not a misprint. You can indeed prove that       f    ′   takes every value between       f    ′    (  a  ) and       f    ′    (  b  ). You cannot, however, assume that       f    ′   is continuous. A standard example is   f  (  x  )  =      x    2    sin  ⁡  (  1      /    x  ) when   x  ≠  0, and 0 otherwise. This function is differentiable at 0 but the derivative isn&#039;t continuous at it.To prove the problem you have, consider the function   g  (  x  )  =  f  (  x  )  −  λ  x for any   λ  ∈  (      f    ′    (  a  )  ,      f    ′    (  b  )  ). What do you know about       g    ′    (  a  ),       g    ′    (  b  )? What do you conclude about   g in the interval   [  a  ,  b  ]?

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