Condition (1):

Since, f(x)=(x-1)(x-2)² is a polynomial and we know every polynomial function is continuous for all xϵR.

⇒ f(x)= (x-1)(x-2)² is continuous on [1,2].

Condition (2):

Here, f’(x)= (x-2)²+2(x-1)(x-2) which exist in [1,2].

So, f(x)= (x-1)(x-2)² is differentiable on (1,2).

Condition (3):

Here, f(1)= (1-1)(1-2)²=0

And f(2)= (2-1)(2-2)²=0

i.e. f(1)=f(2)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one cϵ(1,2) such that f’(c)=0

i.e. (c-2)²+2(c-1)(c-2)=0

(3c-4)(c-2)=0

i.e. c=2 or c=4÷3

Value of

Thus, Rolle’s theorem is satisfied.