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


i.e. c=2 or c=3 or c=18÷7

Value of

Thus, Rolle’s theorem is satisfied.

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