Verify Rolle’s Theorem for the function: f(x) = x2 – 4x + 3 in

1.	Verify Rolle’s Theorem for the function: f(x) = x2 – 4x + 3 in [1, 3]
Answer» We know that (i) f (x) = x² – 4x + 3 is a polynomial which is continuous for all x ϵ R Hence, f (x) = x² – 4x + 3 is continuous on [1, 3] (ii) f’(x) = 2x – 4 exist in [1, 3] Hence, f (x) = x² – 4x + 3 is differentiable on (1, 3) (iii) We know that f (1) = (1)² – 4(1) + 3 = 0 Similarly f (3) = 3² – 4(3) + 3 = 0 Here f (1) = f(3) The conditions of Rolle’s Theorem are satisfied. There exist at least one c ϵ (1, 3) where f’(c) = 0 2c – 4 = 0 which gives c = 4/2 = 2 We know that value of c = 2 ϵ (1, 3) Therefore, Rolle’s Theorem is satisfied.

Verify Rolle’s Theorem for the function: f(x) = x2 – 4x + 3 in [1, 3]

Answer»

We know that

(i) f (x) = x² – 4x + 3 is a polynomial which is continuous for all x ϵ R

Hence, f (x) = x² – 4x + 3 is continuous on [1, 3]

(ii) f’(x) = 2x – 4 exist in [1, 3]

Hence, f (x) = x² – 4x + 3 is differentiable on (1, 3)

(iii) We know that

f (1) = (1)² – 4(1) + 3 = 0

Similarly f (3) = 3² – 4(3) + 3 = 0

Here f (1) = f(3)

The conditions of Rolle’s Theorem are satisfied.

There exist at least one c ϵ (1, 3) where f’(c) = 0

2c – 4 = 0 which gives c = 4/2 = 2

We know that value of c = 2 ϵ (1, 3)

Therefore, Rolle’s Theorem is satisfied.