Determine whether Rolle's Theorem can be applied to the function on the closed interval .If Rolle's Theorem can be applied,find all numbers c in the open interval such that .

Question

Determine whether Rolle's Theorem can be applied to the function  on the closed interval  .If Rolle's Theorem can be applied,find all numbers c in the open interval  such that  . ​

Quizplus · Accepted Answer

To apply Rolle's Theorem, a function must satisfy the following conditions:
1. The function must be continuous on the closed interval [a, b]
2. The function must be differentiable on the open interval (a, b)
3. The function must have the same value at the endpoints of the interval, i.e. f(a) = f(b)

In this case, the function satisfies all three conditions. Therefore, Rolle's Theorem can be applied.

To find the number(s) c in the open interval (a, b) where f'(c) = 0, we need to find the critical points of the function in the open interval.

f(x) = x^3 - 6x^2 + 9x - 2
f'(x) = 3x^2 - 12x + 9
Setting f'(x) = 0, we get:
3x^2 - 12x + 9 = 0
x^2 - 4x + 3 = 0
(x - 1)(x - 3) = 0

Therefore, the critical points of f(x) in the open interval (1, 3) are x = 1 and x = 3. Since f(x) is a polynomial function, it is continuous and differentiable for all x, including the critical points.

Therefore, the answer is A) Rolle's Theorem applies; 1 and 3.

Determine Whether Rolle's Theorem Can Be Applied to the Function