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

1.	Verify Rolle’s theorem for each of the following functions:f(x) = x2 - 4x+3 in [1, 3]
Answer» Condition (1): Since, f(x) = x²- 4x+3 is a polynomial and we know every polynomial function is continuous for all x ϵ R. ⇒ f(x) = x²- 4x+3 is continuous on [1,3]. Condition (2): Here, f’(x) = 2x - 4 which exist in [1,3]. So, f(x) = x²- 4x+3 is differentiable on (1,3). Condition (3): Here, f(1) = (1)²- 4(1)+3 = 0 And f(3) = (3)²- 4(3)+3 = 0 i.e. f(1) = f(3) Conditions of Rolle’s theorem are satisfied. Hence, there exist at least one c ϵ (1,3) such that f’(c) = 0 i.e. 2c - 4 = 0 i.e. c = 2 Value of c = 2 ϵ (1,3) Thus, Rolle’s theorem is satisfied.

Verify Rolle’s theorem for each of the following functions:f(x) = x2 - 4x+3 in [1, 3]

Answer»

Condition (1):

Since, f(x) = x²- 4x+3 is a polynomial and we know every polynomial function is continuous for all x ϵ R.

⇒ f(x) = x²- 4x+3 is continuous on [1,3].

Condition (2):

Here, f’(x) = 2x - 4 which exist in [1,3].

So, f(x) = x²- 4x+3 is differentiable on (1,3).

Condition (3):

Here, f(1) = (1)²- 4(1)+3 = 0

And f(3) = (3)²- 4(3)+3 = 0

i.e. f(1) = f(3)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one c ϵ (1,3) such that f’(c) = 0

i.e. 2c - 4 = 0

i.e. c = 2

Value of c = 2 ϵ (1,3)

Thus, Rolle’s theorem is satisfied.