Answer :

First, let us write the conditions for the applicability of Rolle’s theorem:


For a Real valued function ‘f’:


a) The function ‘f’ needs to be continuous in the closed interval [a,b].


b) The function ‘f’ needs differentiable on the open interval (a,b).


c) f(a) = f(b)


Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.


Given function is:


f(x) = x3 + bx2 + cx, xϵ[1,2]


According to the problem the Rolle’s theorem holds for the function ‘f’ at .


We can say that .


Let’s find the derivative of f(x)




f’(x) = 3x2 + 2bx + c


We have





8b + 3c = – 16 ...... (1)


We also have f(1) = f(2)


(1)3 + b(1)2 + c(1) = (2)3 + b(2)2 + c(2)


1 + b(1) + c = 8 + b(4) + 2c


3b + c = – 7 ......(2)


On solving (1) and (2), we get


b = – 5 and c = 8


The values of b and c is – 5 and 8.


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