# Rolle's Theorem

Rolle's Theorem. Suppose following three condition hold for function y=f(x):

1. function is defined and continuous on closed interval [a,b];
2. exists finite derivative f'(x) on interval (a,b);
3. f(a)=f(b).

then there exists point c (a<c<b) such that f'(c)=0. Geometrically Rolle's theorem means the following: if f(a)=f(b) then there exists point c at which tangent line is horizontal.

Note, that all three conditions are needed.

For function f(x)=x-[x] on interval [0,1] first condition doesn't hold, because it is not continuous at x=1. And f'(x)=1 on (0,1), so there are no point c from (0,1) such that f'(c)=0.

For function f(x)={(x if 0<=x<=1/2),(1-x if 1/2<=x<=1):} second condition doesn't hold, because derivative doesn't exist at x=1/2. Aso f'(x)=1 on (0,1/2) and f'(x)=-1 on (1/2,1), so there are no point c from (0,1) such that f'(c)=0.

For function f(x)=x on interval [0,1] third condition doesn't hold, f(0)!=f(1). And f'(x)=1 everywhere, so there are no point c from (0,1) such that f'(c)=0.