text: # Analysis derivative proof
  
  ## Homework Statement
  
  Let f : (a, b)--> R be differentiable on (a,b), and assume that f'(x) unequal 1 for all x in (a,b).
  Show that there is at most one point c in (a,b) satsifying f(c) = c.
  
  ## The Attempt at a Solution
  
  I think that We need to use the mean value theorem for this problem.
  
  This is what I know, but I'm not sure what I should use for my proof:
  If we let c be n (a,b)
  By definition f'(c)=lim: x-->c (f(x)-f(c)/x-c) assuming that f(x) is differentiable at c.
  
  Also, the mean value theorem states that f[a,b]-->R is continuous on [a.b] and differentiable on (a,b), then there exists a point c in (a,b) so that f'(c)=f(b)-f(a)/b-a
  
  Any help/hints would be great!
explanation: The document is a homework problem and solution attempt related to a proof involving derivatives and the mean value theorem. It contains a question (the homework statement) and an incomplete attempt at a solution.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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distilabel_metadata.role: system
distilabel_metadata.content: # Analysis derivative proof
  
  ## Homework Statement
  
  Let f : (a, b)--> R be differentiable on (a,b), and assume that f'(x) unequal 1 for all x in (a,b).
  Show that there is at most one point c in (a,b) satsifying f(c) = c.
  
  ## The Attempt at a Solution
  
  I think that We need to use the mean value theorem for this problem.
  
  This is what I know, but I'm not sure what I should use for my proof:
  If we let c be n (a,b)
  By definition f'(c)=lim: x-->c (f(x)-f(c)/x-c) assuming that f(x) is differentiable at c.
  
  Also, the mean value theorem states that f[a,b]-->R is continuous on [a.b] and differentiable on (a,b), then there exists a point c in (a,b) so that f'(c)=f(b)-f(a)/b-a
  
  Any help/hints would be great!
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a homework problem and solution attempt related to a proof involving derivatives and the mean value theorem. It contains a question (the homework statement) and an incomplete attempt at a solution.
  
  Label: question-and-answer