Condition (1):

Since, f(x)=(x-1)(2x-3) is a polynomial and we know every polynomial function is continuous for all xϵR.

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

Condition (2):

Here, f’(x)= (2x-3)+ 2(x-1) which exist in [1,3].

So, f(x)= (x-1)(2x-3) is differentiable on (1,3).

Condition (3):

Here, f(1)= (1-1)(2(1)-3)=0

And f(5)= (3-1)(2(3)-3)=6

i.e. f(1)≠f(3)

Condition (3) of Rolle’s theorem is not satisfied.

So, Rolle’s theorem is not applicable.