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.