Q. 3 N

Verify Rolle’s theorem for each of the following functions on the indicated intervals :


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:



We know that sine function is continuous and differentiable over R.


Let’s check the values of function ‘f’ at the extremums,



f(0) = 0 – 4(0)


f(0) = 0






.


We got . So, there exists a cϵ such that f’(c) = 0.


Let’s find the derivative of function ‘f.’







We have f’(c) = 0





We know






Rolle’s theorem is verified.


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