Verify Rolle’s theorem for each of the following functions:f(x) = x3

1.	Verify Rolle’s theorem for each of the following functions:f(x) = x3 - 7x2+16x - 12 in [2, 3]
Answer» Condition (1): Since, f(x) = x³- 7x²+16x -12 is a polynomial and we know every polynomial function is continuous for all x ϵ R. ⇒ f(x) = x³- 7x²+16x -12 is continuous on [2,3]. Condition (2): Here, f’(x) = 3x²- 14x+16 which exist in [2,3]. So, f(x) = x³- 7x²+16x -12 is differentiable on (2,3). Condition (3): Here, f(2) = 2³- 7(2)²+16(2) -12 = 0 And f(3) = 3³- 7(3)²+16(3) -12 = 0 i.e. f(2) = f(3) Conditions of Rolle’s theorem are satisfied. Hence, there exist at least one c ϵ (2,3) such that f’(c) = 0 i.e. 3c²- 14c+16 = 0 i.e. (c - 2)(3c - 7) = 0 i.e. c = 2 or c = 7 ÷ 3 Value of c = 7/3 ϵ (2, 3) Thus, Rolle’s theorem is satisfied.

Verify Rolle’s theorem for each of the following functions:f(x) = x3 - 7x2+16x - 12 in [2, 3]

Answer»

Condition (1):

Since, f(x) = x³- 7x²+16x -12 is a polynomial and we know every polynomial function is continuous for all x ϵ R.

⇒ f(x) = x³- 7x²+16x -12 is continuous on [2,3].

Condition (2):

Here, f’(x) = 3x²- 14x+16 which exist in [2,3].

So, f(x) = x³- 7x²+16x -12 is differentiable on (2,3).

Condition (3):

Here, f(2) = 2³- 7(2)²+16(2) -12 = 0

And f(3) = 3³- 7(3)²+16(3) -12 = 0

i.e. f(2) = f(3)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one c ϵ (2,3) such that f’(c) = 0

i.e. 3c²- 14c+16 = 0

i.e. (c - 2)(3c - 7) = 0

i.e. c = 2 or c = 7 ÷ 3

Value of c = 7/3 ϵ (2, 3)

Thus, Rolle’s theorem is satisfied.