Q. 3 O5.0( 2 Votes )

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

f(x) = 4sin x on [0, π]

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) = 4sinx on [0,]


We that sine function is continuous and differentiable over R.


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


f(0) = 4sin(0)


f(0) = 40


f(0) = 1


f() = 4sinπ


f() = 40


f() = 1


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


Let’s find the derivative of ‘f’





We have f’(c) = 0


4sinclog4cosc = 0


cosc = 0



Rolle’s theorem is verified.


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