Verify Rolle’s theorem for each of the following functions:f(x) = x2

1.	Verify Rolle’s theorem for each of the following functions:f(x) = x2 on [-1, 1]
Answer» Condition (1): Since, f(x) = x² is a polynomial and we know every polynomial function is continuous for all x ϵ R. ⇒ f(x) = x² is continuous on [-1,1]. Condition (2): Here, f’(x) = 2x which exist in [-1,1]. So, f(x) = x² is differentiable on (-1,1). Condition (3): Here, f(-1) = (-1)² = 1 And f(1) = 1¹ = 1 i.e. f(-1) = f(1) Conditions of Rolle’s theorem are satisfied. Hence, there exist at least one c ϵ (-1,1) such that f’(c) = 0 i.e. 2c = 0 i.e. c = 0 Value of c = 0 ϵ (-1,1) Thus, Rolle’s theorem is satisfied.

Verify Rolle’s theorem for each of the following functions:f(x) = x2 on [-1, 1]

Answer»

Condition (1):

Since, f(x) = x² is a polynomial and we know every polynomial function is continuous for all x ϵ R.

⇒ f(x) = x² is continuous on [-1,1].

Condition (2):

Here, f’(x) = 2x which exist in [-1,1].

So, f(x) = x² is differentiable on (-1,1).

Condition (3):

Here, f(-1) = (-1)² = 1

And f(1) = 1¹ = 1

i.e. f(-1) = f(1)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one c ϵ (-1,1) such that f’(c) = 0

i.e. 2c = 0

i.e. c = 0

Value of c = 0 ϵ (-1,1)

Thus, Rolle’s theorem is satisfied.