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

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

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

Answer»

We know that

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

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

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

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

(iii) We know that

f (-π/2) = cos (-π/2) = 0

Similarly f (π/2) = cos (π/2) = 0

Here f (-π/2) = f (π/2)

The conditions of Rolle’s Theorem are satisfied.

There exist at least one c ϵ (-π/2, π/2) where f’(c) = 0

– sin c = 0

Which gives c = 0

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

Therefore, Rolle’s Theorem is satisfied.