Definition of function Rolle:

A function f: [a, b]->R (a<b) is called function Rolle if she is continue on compact interval [a,b] and derivative on the open interval (a, b).

One problem:
It is a function f: [a, b]->R function Rolle. You must demonstrate that there
is at least point c in the interval (a, b) so as f'(c)=(f(a)-f(b))/(c-b).

Demonstration: f is continue on [a, b] and derivative on (a, b).

We are looking for one function h (x) so as h'(c)=0 give us relation f'(c)=(f(a)-f(c))/(c-b).

f(a)-f(c)=f'(c)*(c-b) => f'(c)*(c-b)-f(a)+f(c)=h'(c) =>
h'(x)=f'(x)*(x-b)-f(a)+f(x) =>
h'(x)=(f(x)-f(a))'*(x-b)+f(x)-f(a)*(x-b)'=>
=>h'(x)={[f(x)-f(a)]*(x-b)}'=> h(x)=[f(x)-f(a)]*(x-b).

But f is continue on [a, b] and derivative on (a, b) => h is continue on [a, b] and derivative on (a, b), because he is the product of two functions continues h (a)=0 and h(b)=[f(b)-f(a)]*0=0 => h - verifies the theoreme's Rolle conditions.

So, there is there at least point c in the interval (a, b) so as h'(c)=0.
Then result that h'(x)=f'(x)*(x-b)+f(x)-f(a) =>
h'(c)=f'(c)*(c-b)+f(c)-f(a) =>h'(c)=0 =>
=> f'(c)*(c-b)+f(c)-f(a)=0 => f'(c)=(f(a)-f(c))/(c-b).

Problem written by Camelia Sima
Teacher: Ligia Garlea
"Alex. Papiu Ilarian" High school Dej, Cluj, Romania