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in Continuity and Differentiability by (29.1k points)
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Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = sin 2x on [0, π/2]

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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.

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