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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) = 2 sin x + sin 2x on [0, π]

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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:

⇒ f(x) = 2sinx + sin2x on [0, π]

We know that sine function continuous and differentiable over R.

Let’s check the values of function f at the extremums

⇒ f(0) = 2sin(0) + sin2(0)

⇒ f(0) = 2(0) + 0

⇒ f(0) = 0

⇒ f(π) = 2sin(π) + sin2(π)

⇒ f(π) = 2(0) + 0

⇒ f(π) = 0

We have got f(0) = f(π). So, there exists a c∈(0, π) such that f’(c) = 0.

Let’s find the derivative of function ‘f’.

∴ Rolle’s theorem is verified.

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