Question

Determine if the function f(x)=x√6​−x satisfies the hypotheses of Rolle's Theorem on the interval [0, 6]. If it does, find all numbers c satisfying the conclusion of that theorem.

Answer 1

The function f(x) = x√6 - x does not satisfy all of the hypotheses of Rolle's Theorem over the interval [0, 6] because the function values are not equal at the endpoints of the interval; therefore, there is no need to find a number c in the interval where the derivative of the function is zero.

Step-by-step explanation:

To determine if the function f(x)=x√6​−x satisfies the hypotheses of Rolle's Theorem on the interval [0, 6], we must check three conditions:

1. The function must be continuous on the closed interval [0, 6].
2. The function must be differentiable on the open interval (0, 6).
3. The function must have equal values at the endpoints of the interval.

f(x)=x√6​−x is a polynomial function, thus it is continuous and differentiable on any interval, including [0, 6]. Now let's evaluate the function at the endpoints to ensure the third condition is met:

f(0) = 0√6 - 0 = 0

f(6) = 6√6 - 6 = 6(√6 - 1)

The values of the function at the endpoints are not equal, therefore the third condition of Rolle's Theorem is not met. As a result, Rolle's Theorem cannot be applied to f(x) = x√6​ - x over the interval [0, 6], and it is unnecessary to search for a number c in the interval (0, 6) where the derivative f'(x) = 0.