Answer :

Condition (1):


Since, f(x)=(x-2)4(x-3)3 is a polynomial and we know every polynomial function is continuous for all xϵR.


f(x)= (x-2)4(x-3)3 is continuous on [2,3].


Condition (2):


Here, f’(x)= 4(x-2)3(x-3)3+3(x-2)4(x-3)2 which exist in [2,3].


So, f(x)= (x-2)4(x-3)3 is differentiable on (2,3).


Condition (3):


Here, f(2)= (2-2)4(2-3)3=0


And f(3)= (3-2)4(3-3)3=0


i.e. f(2)=f(3)


Conditions of Rolle’s theorem are satisfied.


Hence, there exist at least one cϵ(2,3) such that f’(c)=0


i.e. 4(c-2)3(c-3)3+3(c-2)4(c-3)2=0


(c-2)3(c-3)2(7c-18)=0


i.e. c=2 or c=3 or c=18÷7


Value of


Thus, Rolle’s theorem is satisfied.


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