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

Question

Examine if the Rolle’s theorem applies to anyone of the following functions: f(x) = [x] for x ϵ [ – 2,2]

Philoid · Accepted Answer

Given function is:

⇒ f(x) = [x] for xϵ[ – 2,2]

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

Here in the interval xϵ[ – 2,1], the function has to be Right continuous at x = 2 and left continuous at x = 2.

Right Hand Limit:

⇒

⇒ where h>0.

⇒

⇒ ......(1)

Left Hand Limit:

⇒

⇒ , where h>0

⇒

⇒ ......(2)

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

∴ Rolle’s theorem is not applicable for the function f in the interval [ – 2,2].