Examine if Rolle's theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle's theorem from these examples?
(i) f(x) $=$ [x] for x $\in$ [5, 9]
(ii) f(x) $=$ [x] for x$\in$[-2, 2]
(iii) f(x) $=$ ${x^2} -$1 for x$\in$ [1, 2].
(i) Being greatest integer function, the given function is not differentiable and continuous. Hence, Rolle's theorem is not applicable.
(ii) Being greatest integer function, the given function is not differentiable and continuous. Hence, Rolle's theorem is not applicable.
(iii) $f(x) = {x^2} - 1,x \in$[1, 2] is a polynomial function. So,
(a) It is continuous in [1, 2]
(b) It is derivable in (1, 2) and
(c) $f(1) = {(1)^2} - 1 = 1 - 1 = 0$
$f(2) = {(2)^2} - 1 = 4 - 1 = 3$
As $f(1) \ne f(2)$, hence Rolle's theorem is not applicable.
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