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 - x - 12 in [-3, 4]
Answer» Condition (1): Since, f(x) = x²- x -12 is a polynomial and we know every polynomial function is continuous for all x ϵ R. ⇒ f(x) = x²- x -12 is continuous on [-3,4]. Condition (2): Here, f’(x) = 2x -1 which exist in [-3,4]. So, f(x) = x²- x -12 is differentiable on (-3,4). Condition (3): Here, f(-3) = (-3)² - 3 -12 = 0 And f(4) = 4²- 4 -12 = 0 i.e. f(-3) = f(4) Conditions of Rolle’s theorem are satisfied. Hence, there exist at least one c ϵ (-3,4) such that f’(c) = 0 i.e. 2c-1 = 0 i.e. c = 1/2 Value of c = 1/2 ϵ (-3,4) Thus, Rolle’s theorem is satisfied.

Verify Rolle’s theorem for each of the following functions:f(x) = x2 - x - 12 in [-3, 4]

Answer»

Condition (1):

Since, f(x) = x²- x -12 is a polynomial and we know every polynomial function is continuous for all x ϵ R.

⇒ f(x) = x²- x -12 is continuous on [-3,4].

Condition (2):

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

So, f(x) = x²- x -12 is differentiable on (-3,4).

Condition (3):

Here, f(-3) = (-3)² - 3 -12 = 0

And f(4) = 4²- 4 -12 = 0

i.e. f(-3) = f(4)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one c ϵ (-3,4) such that f’(c) = 0

i.e. 2c-1 = 0

i.e. c = 1/2

Value of c = 1/2 ϵ (-3,4)

Thus, Rolle’s theorem is satisfied.