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) = x^3−x^2−20x + 7, [0, 5]

nevermind i figured it out already.

To verify that the function satisfies the three hypotheses of Rolle's Theorem, we need to check the following:

1. Continuity: We need to make sure that the function is continuous on the closed interval [0, 5].

2. Differentiability: We need to ensure that the function is differentiable on the open interval (0, 5).

3. Endpoint values: We need to check if the values of the function at the endpoints of the interval are equal.

Let's go through each of these steps:

1. Continuity: To check continuity, we need to ensure that the function is defined and there are no jumps, holes, or vertical asymptotes in the interval [0, 5].

The function f(x) = x^3 - x^2 - 20x + 7 is a polynomial function, and all polynomial functions are continuous for all real numbers. Therefore, our function is continuous on the interval [0, 5].

2. Differentiability: To check differentiability, we need to ensure that the derivative of the function exists and is continuous on the open interval (0, 5).

To find the derivative of f(x), we differentiate term by term using the power rule:

f'(x) = 3x^2 - 2x - 20.

The derivative is another polynomial function, and all polynomial functions are differentiable for all real numbers. Therefore, our function is differentiable on the open interval (0, 5).

3. Endpoint values: We need to check the values of the function at the endpoints of the interval [0, 5].

For f(x), we can find:

f(0) = (0)^3 - (0)^2 - 20(0) + 7 = 7.

f(5) = (5)^3 - (5)^2 - 20(5) + 7 = -133.

The values of the function at the endpoints are f(0) = 7 and f(5) = -133.

Now, to find the numbers c that satisfy the conclusion of Rolle's Theorem, we need to find the points within the open interval (0, 5) where the derivative is zero.

Setting f'(x) = 0, we have:

3x^2 - 2x - 20 = 0.

Solving this quadratic equation, we find two possible solutions:

x = -2, x = (10/3).

The numbers c that satisfy the conclusion of Rolle's Theorem are -2 and (10/3).

Therefore, the answers are -2, (10/3).