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प्रश्न
Examine if Rolle's theorem is applicable to any one of the following functions.
(i) f (x) = [x] for x ∈ [5, 9]
(ii) f (x) = [x] for x ∈ [−2, 2]
Can you say something about the converse of Rolle's Theorem from these functions?
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उत्तर
By Rolle’s theorem, for a function f : [a , b] →R , if
(a) f is continuous on [a, b],
(b) f is differentiable on (a, b) and
(c) f (a) = f (b),
then there exists some c ∈ (a, b) such that f'(c) = 0 .
Therefore, Rolle’s theorem is not applicable to those functions that do not satisfy any of the three conditions of the hypothesis
(i) f(x) = [x] for x ∈ [ 5 , 9 ]
It is evident that the given function f (x) is not continuous at every integral point.
In particular, f(x) is not continuous at x = 5 and x = 9.
Thus, f (x) is not continuous on [5, 9].
Also , f (5) = [5] = 5 and f (9) = [9] = 9
∴ f (5) ≠ f (9)
The differentiability of f on (5, 9) is checked in the following way.
Let n be an integer such that n ∈ (5, 9).
The left hand limit of f at x = n is,
`lim_(h ->o)(f (n + h) - f (n)\)/h = lim_(h->o) ([n + h ] - [n])/h) = lim_(h->o)(n- 1- n)/h = lim_(h ->o)(-1)/h =oo`
The right hand limit of f at x = n is,
`lim_(h->o) (f (n +h )- f (n))/h = lim _(h->o)([n +h] - [n])/h = lim_(h->o)(n-n)/h = lim _(h->o) 0 = 0 `
Since the left and the right hand limits of f at x = n are not equal, f is not differentiable at x = n.
Thus, f is not differentiable on (5, 9).
It is observed that f does not satisfy all the conditions of the hypothesis of Rolle’s theorem.
Hence, Rolle’s theorem is not applicable on f (x) for x ∈ [5 , 9] .
(ii) f (x) = [x] for x ∈ [-2 , 2]
It is evident that the given function f (x) is not continuous at every integral point.
In particular, f(x) is not continuous at x = −2 and x = 2.
Thus, f (x) is not continuous on [−2, 2].
Also, f (-2) = [-2] = -2 and f (2) = [2] = 2
∴ f (-2) ≠ f (2)
The differentiability of f on (−2, 2) is checked in the following way.
Let n be an integer such that n ∈ (−2, 2).
The left hand limit of f at x = n is .
`lim_(h->o) (f (n + h) -f (n))/ h = lim_(h->o) ([n +h]- [n])/h = lim_(h->o)(n - 1- n)/h = lim_(h->o) (-1)/ h =oo`
The right hand limit of f at x = n is ,
`lim_(h->o) (f(n+h) -f (n))/h = lim_(h->o)([n+h] - [n])/h) lim_(h->o)(n-n)/h = lim_(h->o)(-1)/h = oo`
Since the left and the right hand limits of f at x = n are not equal, f is not differentiable at x = n.
Thus, f is not differentiable on (−2, 2).
It is observed that f does not satisfy all the conditions of the hypothesis of Rolle’s theorem.
Hence, Rolle’s theorem is not applicable on f(x)= [x] for x ∈ [ -2 , 2 ].
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