proove that suppose 'f' is continious on [a,b] , f'(x) exists for each
x in (a,b) and f(a)= f(b) , then there is a 'c' in  (a,b) suchthat
f'(c)=0.
Hint: the tangent line to f at (c,f(c)) is horizontal.
      please give me the detailed proof.

Hi maria2002!
The theorem you're trying to prove is called the "Rolle's theorem",
and a very important corollary of it is that, at a local extremum
(either maximum or minimum) of a continuous differentiable function,
the derivative is equal to 0. The theorem itself can be found at

PlanetMath: Rolle's Theorem
http://planetmath.org/?op=getobj&from=objects&id=422

and the proof is at

PlanetMath: Proof of Rolle's Theorem
http://planetmath.org/?op=getobj&from=objects&id=2947


Google search strategy
"rolle's theorem" proof
://www.google.com/search?sourceid=navclient&ie=UTF-8&oe=UTF-8&q=%22rolle%27s+theorem%22+proof


I hope this helps! If you have any doubt regarding my answer, please
don't hesitate to request a clarification before rating it. Otherwise
I await your rating and final comments.

Best wishes!
elmarto