Examine whether Rolles Theorem is applicable to the following functions for the interval in which they are defined.




SOLUTION 
Below is a table of intervals of functions which will be useful 
 in interpreting any given interval. 






(i) To examine Rolles Theorem in this function we first interprete the given interval 0<x<1 using the table above,  this  can be interpreted in terms of an open interval (0,1).
Now we check if Rolles Theorem is applicable f(0)=-1, f(1)=0. 
Since f(0)≠f(1) then the function doest  not satisfy Rolles Theorem, although the function is continuous on the interval. 
(ii) we first interprete the given interval using the table above 0≤x≤2,this interval is a closed interval [0,2],the given  function is differentiable in this interval. Now let's check If Rolles Theorem is applicable.


 








Hence f(0)=f(2)=4 which satisfies Rolles Theorem. 
Let's check for the stationary point f'(c)=0, to do this we first take the derivative of the function. 









Hence the stationary point cannot be obtained.