Examine if Rolles theorem is applicable to any of the following functions. Can you say something about the converse of Rolles theorem from these example?(i) `f(x)=[x]`for `x in [5,9]`(ii) `f(x) = [x]`for `x in [-2,2]`(iii) `f(x)=x^

Explanation

(i) `f(x) =[x], " " x in[5, 9]`
`f(x)` is discontinuous on integral points.
Hence, Rolle's theorem is not applicable.
(ii) `f(x)=[x], " " x in [-2, 2]`
`f(x)` is discontinuous on intergral points.
Hence, Rolle's theorem is not applicable.
(iii) `f(x)=x^(2)-1, " " x in [1, 2]`
`f(x)` is a polynomial function.
` :. f(x)` is continuous in [1, 2] and differentiate in (1, 2)
`f(1) = 1^(2)-1=0`
and `f(2)=2^(2)-1=3`
` :. f(1) ne f(2)`
Hence, Rolle's theorem is not applicable.
The converse of Rolle's theorem is not true.