define and prove rolles theorem

If areal-valued functionfiscontinuouson aclosed interval[a,b],differentiableon theopen interval(a,b), andf(a)=f(b), then there exists acin the open interval (a,b) such that df(x)/dx at x=c is zero.

Proof:

It can be claimed, since the function is continuous and its value at end points is equal, so it must attains a maximum in between at some point c, which imply that its derviative should be zero at that point.