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:
\(\Rightarrow f(x)=sin\, 2x\) on \([0, \frac{\pi}{2}]\)
We know that sine function is continuous and differentiable on R.
Let’s find the values of function at extremum,
⇒ f(0) = sin2(0)
⇒ f(0) = sin0
⇒ f(0) = 0
We got f(0) = f\((\frac{\pi}{2}),\) so there exist a c∈\((0,\frac{\pi}{2})\) such that f’(c) = 0.
Let’s find the derivative of f(x)
∴ Rolle’s theorem is verified.