Verify Rolle’s Theorem for the function: f(x) = x3 + 3x2 – 24x

1.	Verify Rolle’s Theorem for the function: f(x) = x3 + 3x2 – 24x – 80 in [- 4, 5]
Answer» We know that (i) f (x) = x³ + 3x² – 24x – 80 is a polynomial which is continuous for all x ϵ R Hence, f (x) = x³ + 3x² – 24x – 80 is continuous on [- 4, 5] (ii) f’(x) = 3x² + 6x – 24 exist in [- 4, 5] Hence, f (x) = x³ + 3x² – 24x – 80 is differentiable on (- 4, 5) (iii) We know that f (- 4) = (- 4)³ + 3 (4)² – 24(4) – 80 = 0 Similarly f (5) = 5³ + 3(5)² – 24 (5) – 80 = 0 Here f (- 4) = f (5) The conditions of Rolle’s Theorem are satisfied. There exist at least one c ϵ (- 4, 5) where f’(c) = 0 3c² + 6c – 24 = 0 Which gives c = – 4 or 2 We know that value of c = 2 ϵ (- 4, 5) Therefore, Rolle’s Theorem is satisfied.

Verify Rolle’s Theorem for the function: f(x) = x3 + 3x2 – 24x – 80 in [- 4, 5]

Answer»

We know that

(i) f (x) = x³ + 3x² – 24x – 80 is a polynomial which is continuous for all x ϵ R

Hence, f (x) = x³ + 3x² – 24x – 80 is continuous on [- 4, 5]

(ii) f’(x) = 3x² + 6x – 24 exist in [- 4, 5]

Hence, f (x) = x³ + 3x² – 24x – 80 is differentiable on (- 4, 5)

(iii) We know that

f (- 4) = (- 4)³ + 3 (4)² – 24(4) – 80 = 0

Similarly f (5) = 5³ + 3(5)² – 24 (5) – 80 = 0

Here f (- 4) = f (5)

The conditions of Rolle’s Theorem are satisfied.

There exist at least one c ϵ (- 4, 5) where f’(c) = 0

3c² + 6c – 24 = 0

Which gives c = – 4 or 2

We know that value of c = 2 ϵ (- 4, 5)

Therefore, Rolle’s Theorem is satisfied.