Question

Examine if Rolle's theorem is applicable to any of the following functions. Can you say something about the converse of Rolle's theorem from these example ?

(i) $f\left(x\right)=\left[x\right]forxϵ[-2,2]$

Answer 1

(i) Given function is f(x)=f(x),x $ϵ[5,9]$

Since, f(x) is neither continuous nor differentable at integral point i.e., at -2.0,1,2

Thus, f(x) is not continuous on [-2,2].

So, f is not deirvable on (-2,2).

Moreover, $f(-2)=[-2]=-2\ne 2=f\left(2\right)$ f.e., $f\left(5\right)\ne f\left(9\right)$.

Hence, Rolle's theorem is not applicable to f(x) in the given interval.

It may be noted that f'(x)=0 for all non-integral points in [-2,2].