Q. 7 A5.0( 2 Votes )
Examine if the Rolle’s theorem applies to anyone of the following functions:
f(x) = [x] for xϵ[5,9]
Answer :
First, let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
Given function is:
⇒ f(x) = [x] for xϵ[5,9]
Let us check the continuity of the function ‘f’.
Here in the interval xϵ[5,9], the function has to be Right continuous at x = 5 and left continuous at x = 5.
Right Hand Limit:
⇒ 
⇒ 
where h>0.
⇒ 
⇒ 
......(1)
Left Hand Limit:
⇒ 
⇒ 
, where h>0
⇒ 
⇒ 
......(2)
From (1) and (2), we can clearly see that the limits are not same so, the function is not continuous in the interval [5,9].
∴ Rolle’s theorem is not applicable for the function f in the interval [5,9].
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