No...

After all, consider the function y=x on [1,2]. It is differentiable at every point of the interval, but doesn't have any place where it has zero derivative.

Analysis WebNotes by John Lindsay Orr.
Comments to the author: jorr@math.unl.edu

All contents copyright (C) 1996 John L. Orr
University of Nebraska--Lincoln
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Last modified: May 1996