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 fuction is:
⇒ f(x) = sin2x on
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 , so there exist a such that f’(c) = 0.
Let’s find the derivative of f(x)
⇒ f’(x) = 2cos2x
We have f’(c) = 0,
⇒ 2cos2c = 0
∴ Rolle’s theorem is verified.
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