Q. 1 A4.7( 3 Votes )
Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = 3 + (x – 2)2/3 on [1, 3]
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.
(i) Given function is:
⇒ 
on [1,3]
Let us check the differentiability of the function f(x).
Let’s find the derivative of f(x),
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Let’s the differentiability at the value of x = 2
⇒ 
⇒ 
⇒ 
⇒ 
∴ f is not differentiable at x = 2, so it is not differentiable in the closed interval (1,3).
So, Rolle’s theorem is not applicable for the function f on the interval [1,3].
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