Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = x3 − x2 − 2x + 1, [0, 2]

Step-by-step explanation:

Rolle's theorem states that if f is a continuous function defined on a closed interval [a, b] differentiable on the open interval (a, b) and f (a) = f (b), then:

There is at least one point c in the open interval (a, b) such that f '(c) = 0

Given a function:

and a interval:

Is f(x) continuous over [0,2]?

Yes, it is, because the domain of this function is

Is f'(x) differentiable on the open interval (0,2) ?

First, let's find f'(x):

Therefore f'(x) is differentiable on the open interval (0,2) because the domain of f'(x) is

Is f(0)=f(2) ?

Hence:

Now, we have verified that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. So, let's find all numbers c that satisfy the conclusion of Rolle's Theorem:

Let's find the roots using the quadratic equation:

Since:

The value of c that satisfies the conclusion of Rolle's Theorem is:

.

step-by-step explanation: