Verify Rolle’s Theorem for the function: f(x) = cos 2x in [0, π]

1.	Verify Rolle’s Theorem for the function: f(x) = cos 2x in [0, π]
Answer» We know that (i) f (x) = cos 2x is a trigonometric function which is continuous Hence, f (x) = cos 2x is continuous on [0, π] (ii) f’(x) = – 2sin 2x exist in [0, π] Hence, f (x) = cos 2x is differentiable on (0, π) (iii) We know that f (0) = cos 2(0) = 0 Similarly f (π) = cos 2(π) = 0 Here f (0) = f (π) The conditions of Rolle’s Theorem are satisfied. There exist at least one c ϵ (0, π) where f’(c) = 0 – 2sin 2c = 0 Which gives 2c = π We know that value of c = π/2 ϵ (0, π) Therefore, Rolle’s Theorem is satisfied.

Answer»

We know that

(i) f (x) = cos 2x is a trigonometric function which is continuous

Hence, f (x) = cos 2x is continuous on [0, π]

(ii) f’(x) = – 2sin 2x exist in [0, π]

Hence, f (x) = cos 2x is differentiable on (0, π)

(iii) We know that

f (0) = cos 2(0) = 0

Similarly f (π) = cos 2(π) = 0

Here f (0) = f (π)

The conditions of Rolle’s Theorem are satisfied.

There exist at least one c ϵ (0, π) where f’(c) = 0

– 2sin 2c = 0

Which gives 2c = π

We know that value of c = π/2 ϵ (0, π)

Therefore, Rolle’s Theorem is satisfied.

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