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) = sin3x on [0, π]
We know that sine function is continuous and differentiable on R.
Let’s find the values of function at extremum,
⇒ f(0) = sin3(0)
⇒ f(0) = sin0
⇒ f(0) = 0
⇒ f(π) = sin3(π)
⇒ f(π) = sin(3π)
⇒ f(π) = 0
We got f(0) = f(π), so there exist a c∈(0, π) such that f’(c) = 0.
Let’s find the derivative of f(x)
∴ Rolle’s theorem is verified.