Verify Rolle’s Theorem for the function: f(x) = x(x – 4)2 in [

1.	Verify Rolle’s Theorem for the function: f(x) = x(x – 4)2 in [0, 4]
Answer» We know that (i) f (x) = x (x – 4)² is a polynomial which is continuous for all x ϵ R Hence, f (x) = x (x – 4)² is continuous on [0, 4] (ii) f’(x) = (x – 4)² + 2x (x – 4) exist in [0, 4] Hence, f (x) = x (x – 4)² is differentiable on (0, 4) (iii) We know that f (0) = 0(0 – 4)² = 0 Similarly f (4) = 4 (4 – 4)² = 0 Here f (0) = f(4) The conditions of Rolle’s Theorem are satisfied. There exist at least one c ϵ (0, 4) where f’(c) = 0 (c – 4)² + 2c (c – 4) = 0 It can be written as (c – 4) (3c – 4) = 0 Here c = 4 or c = 3/4 We know that value of c = 3/4 ϵ (0, 4) Therefore, Rolle’s Theorem is satisfied.

Verify Rolle’s Theorem for the function: f(x) = x(x – 4)2 in [0, 4]

Answer»

We know that

(i) f (x) = x (x – 4)² is a polynomial which is continuous for all x ϵ R

Hence, f (x) = x (x – 4)² is continuous on [0, 4]

(ii) f’(x) = (x – 4)² + 2x (x – 4) exist in [0, 4]

Hence, f (x) = x (x – 4)² is differentiable on (0, 4)

(iii) We know that

f (0) = 0(0 – 4)² = 0

Similarly f (4) = 4 (4 – 4)² = 0

Here f (0) = f(4)

The conditions of Rolle’s Theorem are satisfied.

There exist at least one c ϵ (0, 4) where f’(c) = 0

(c – 4)² + 2c (c – 4) = 0

It can be written as

(c – 4) (3c – 4) = 0

Here c = 4 or c = 3/4

We know that value of c = 3/4 ϵ (0, 4)

Therefore, Rolle’s Theorem is satisfied.