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

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

Verify Rolle’s theorem for each of the following functions:f(x) = x(x - 4)2 in [0, 4]

Answer»

Condition (1):

Since, f(x) = x(x - 4)² is a polynomial and we know every polynomial function is continuous for all x ϵ R.

⇒ f(x) = x(x - 4)² is continuous on [0,4].

Condition (2):

Here, f’(x) = (x - 4)²+2x(x - 4) which exist in [0,4].

So, f(x) = x(x - 4)² is differentiable on (0,4).

Condition (3):

Here, f(0) = 0(0 - 4)² = 0

And f(4) = 4(4 - 4)²= 0

i.e. f(0) = f(4)

Conditions of Rolle’s theorem are satisfied.

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

i.e. (c - 4)²+2c(c - 4) = 0

i.e. (c - 4)(3c - 4) = 0

i.e. c = 4 or c = 3 ÷ 4

Value of c = 3/4 ϵ (0, 4)

Thus, Rolle’s theorem is satisfied.