Verify Rolle’s theorem for each of the following functions:Show that

1.	Verify Rolle’s theorem for each of the following functions:Show that f(x) = x(x - 5)2 satisfies Rolle’s theorem on [0, 5] and that the value of c is (5/3)
Answer» Condition (1): Since, f(x) = x(x - 5)² is a polynomial and we know every polynomial function is continuous for all x ϵ R. ⇒ f(x) = x(x - 5)² is continuous on [0,5]. Condition (2): Here, f’(x) = (x-5)²+ 2x(x - 5) which exist in [0,5]. So, f(x) = x(x - 5)² is differentiable on (0,5). Condition (3): Here, f(0) = 0(0 - 5)²= 0 And f(5) = 5(5 - 5)²= 0 i.e. f(0) = f(5) Conditions of Rolle’s theorem are satisfied. Hence, there exist at least one cϵ(0,5) such that f’(c) = 0 i.e. (c - 5)²+ 2c(c - 5) = 0 i.e.(c - 5)(3c - 5) = 0 i.e. c = 5/3 or c = 5 Value of c = 5/3 ∈ (0, 5) Thus, Rolle’s theorem is satisfied.

Verify Rolle’s theorem for each of the following functions:Show that f(x) = x(x - 5)2 satisfies Rolle’s theorem on [0, 5] and that the value of c is (5/3)

Answer»

Condition (1):

Since, f(x) = x(x - 5)² is a polynomial and we know every polynomial function is continuous for all x ϵ R.

⇒ f(x) = x(x - 5)² is continuous on [0,5].

Condition (2):

Here, f’(x) = (x-5)²+ 2x(x - 5) which exist in [0,5].

So, f(x) = x(x - 5)² is differentiable on (0,5).

Condition (3):

Here, f(0) = 0(0 - 5)²= 0

And f(5) = 5(5 - 5)²= 0

i.e. f(0) = f(5)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one cϵ(0,5) such that f’(c) = 0

i.e. (c - 5)²+ 2c(c - 5) = 0

i.e.(c - 5)(3c - 5) = 0

i.e. c = 5/3 or c = 5

Value of c = 5/3 ∈ (0, 5)

Thus, Rolle’s theorem is satisfied.