Verify Rolle’s theorem for each of the following functions:

Condition (1):

Since, is a polynomial and we know every polynomial function is continuous for all xϵR.

⇒ is continuous on [-1,1].

Condition (2):

Here, which exist in [-1,1].

So, is differentiable on (-1,1).

Condition (3):

Here,

And

i.e.

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one cϵ(-1,1) such that f’(c)=0

i.e.

i.e. c=0

Value of c=0ϵ(-1,1)

Thus, Rolle’s theorem is satisfied.