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:
\(\Rightarrow f(x)=e^xsin\,x\) on [0, π]
We know that exponential and sine functions are continuous and differentiable on R.
Let’s find the values of the function at an extremum,
⇒ f(0) = e0sin(0)
⇒ f(0) = 1 × 0
⇒ f(0) = 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.