Q. 7 A5.0( 2 Votes )

Examine if the Rolle’s theorem applies to anyone of the following functions:

f(x) = [x] for xϵ[5,9]

Answer :

First, let us write the conditions for the applicability of Rolle’s theorem:


For a Real valued function ‘f’:


a) The function ‘f’ needs to be continuous in the closed interval [a,b].


b) The function ‘f’ needs differentiable on the open interval (a,b).


c) f(a) = f(b)


Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.


Given function is:


f(x) = [x] for xϵ[5,9]


Let us check the continuity of the function ‘f’.


Here in the interval xϵ[5,9], the function has to be Right continuous at x = 5 and left continuous at x = 5.


Right Hand Limit:



where h>0.



......(1)


Left Hand Limit:



, where h>0



......(2)


From (1) and (2), we can clearly see that the limits are not same so, the function is not continuous in the interval [5,9].


Rolle’s theorem is not applicable for the function f in the interval [5,9].


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