Question

Discuss the applicability for Rolle’s theorem, when:

f(x) = 2+(x - 1)^2/3 on [0, 2]

Answer 1

Condition (1):

Since, f(x) = 2+(x -1)^2/3 is a polynomial and we know every polynomial function is continuous for all x ϵ R.

⇒ f(x) = 2+(x -1)^2/3 is continuous on [0, 2].

Condition (2):

Here, f'(x) = \(\frac{2}{3(x-1)^{1/3}}\) which does not exist at x = 1 in [0, 2].

f(x) = 2+(x -1)^2/3 is not differentiable on (0, 2).

Condition (2) of Rolle’s theorem is not satisfied.

So, Rolle’s theorem is not applicable.