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 ?
(iii) f(x)=x2−1forx ϵ [5,9].
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 ?
(iii) f(x)=x2−1forx ϵ [5,9].
Given, f(x) = x2−1, Which is a polynomial function. It is continuous and derivable at all x ϵ R.
In particular, f(x) is continuous on [1,2] and derivable on (1,2)
f(1)=12−1=0 and f(2)=22−1=3 i.e., f(1)≠f(2).
∴ Rolle's theorem is not applicable to given function in the given interval. Note that f'(x) = 2x for any x in (1,2)
Conclusion From the above examples, we conclude that the converse of Rolle's theorem does not hold. This means that it conditions fo Rolle's theorem doed theorem are not satified by a function f(x) on [a,b], then f'(x) may or may not vanish at some point in (a,b),
Given, f(x) = x2−1, Which is a polynomial function. It is continuous and derivable at all x ϵ R.
In particular, f(x) is continuous on [1,2] and derivable on (1,2)
f(1)=12−1=0 and f(2)=22−1=3 i.e., f(1)≠f(2).
∴ Rolle's theorem is not applicable to given function in the given interval. Note that f'(x) = 2x for any x in (1,2)
Conclusion From the above examples, we conclude that the converse of Rolle's theorem does not hold. This means that it conditions fo Rolle's theorem doed theorem are not satified by a function f(x) on [a,b], then f'(x) may or may not vanish at some point in (a,b),
