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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) = x(x – 4)2 on [0, 4]

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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) = x(x – 4)2 on [0, 4]

Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R.

Let us find the values at extremums:

⇒ f(0) = 0(0 – 4)2

⇒ f(0) = 0

⇒ f(4) = 4(4 – 4)2

⇒ f(4) = 4(0)2

⇒ f(4) = 0

∴ f(0) = f(4), Rolle’s theorem applicable for function ‘f’ on [0, 4].

Let’s find the derivative of f(x):

\(\Rightarrow f'(x) = \frac{d(x(x-4)^2)}{dx}\)

Differentiating using UV rule,

We have f’(c) = 0 c∈(0,4), from the definition given above.

∴ Rolle’s theorem is verified.

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