Verify Rolle’s theorem for each of the following functions on the indicated intervals :
f(x) = 4sin x on [0, π]
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.