Verify Rolle’s theorem for each of the following functions on the indicated intervals :
f(x) = sin x – sin 2x 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) = sinx – sin2x on [0,
]
We know that sine function is continuous and differentiable over R.
Let’s check the values of the function ‘f’ at the extremums.
⇒ f(0) = sin(0)–sin2(0)
⇒ f(0) = 0 – sin(0)
⇒ f(0) = 0
⇒ f(
) = sin(
) – sin2(
)
⇒ f(
) = 0 – sin(2
)
⇒ f(
) = 0
We got f(0) = f(
). So, there exists a cϵ
such that f’(c) = 0.
Let’s find the derivative of the function ‘f’
⇒ 
⇒ 
⇒ 
⇒ f’(x) = cosx – 2(2cos2x – 1)
⇒ f’(x) = cosx – 4cos2x + 2
We have f’(c) = 0
⇒ cosc – 4cos2c + 2 = 0
⇒ 
⇒ 
⇒ 
We can see that cϵ(0,
)
∴ Rolle’s theorem is verified.