Since, f(x)=x(x-4)2 is a polynomial and we know every polynomial function is continuous for all xϵR.
⇒ f(x)= x(x-4)2 is continuous on [0,4].
Here, f’(x)= (x-4)2+2x(x-4) which exist in [0,4].
So, f(x)= x(x-4)2 is differentiable on (0,4).
And f(4)= 4(4-4)2=0
Conditions of Rolle’s theorem are satisfied.
Hence, there exist at least one cϵ(0,4) such that f’(c)=0
i.e. c=4 or c=3÷4
Thus, Rolle’s theorem is satisfied.
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