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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) = log(x2 + 2) – log 3 on [–1, 1]

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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) = log(x2 + 2) – log3 on [– 1,1]

We know that logarithmic function is continuous and differentiable in its own domain.

We check the values of the function at the extremum,

⇒ f(– 1) = log((– 1)2 + 2) – log3

⇒ f(– 1) = log(1 + 2) – log3

⇒ f(– 1) = log3 – log3

⇒ f(– 1) = 0

⇒ f(1) = log(12 + 2) – log3

⇒ f(1) = log(1 + 2) – log3

⇒ f(1) = log3 – log3

⇒ f(1) = 0

We have got f(– 1) = f(1). So, there exists a c such that c∈(– 1,1) such that f’(c) = 0.

Let’s find the derivative of the function f,

∴ Rolle’s theorem is verified.

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