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`.

rolle theoremGeometrically 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`.