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, π]
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(π) = 2sin(π) + sin2(π)
⇒ f(π) = 2(0) + 0
⇒ f(π) = 0
We have got f(0) = f(π). So, there exists a c∈(0, π) such that f’(c) = 0.
Let’s find the derivative of function ‘f’.
∴ Rolle’s theorem is verified.