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].