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