Question 4.2 Completect: 2 of 6 Determine whether Rolle's Theorem applies to the following function on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. \( g(x)=x^{3}-x^{2}-5 x-3 ;[-1,3] \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. Rolle's Theorem applies and the point(s) guaranteed to exist is/are \( x= \)

To determine whether Rolle's Theorem applies to the function \( g(x) = x^3 - x^2 - 5x - 3 \) on the interval \([-1, 3]\), we need to check the following conditions: 1. The function \( g(x) \) is continuous on the closed interval \([-1, 3]\). 2. The function \( g(x) \) is differentiable on the open interval \((-1, 3)\). 3. The function \( g(x) \) has the same value at the endpoints of the interval, i.e., \( g(-1) = g(3) \). Let's check each condition: 1. The function \( g(x) = x^3 - x^2 - 5x - 3 \) is a polynomial function, which is continuous everywhere. Therefore, it is continuous on the closed interval \([-1, 3]\). 2. The function \( g(x) = x^3 - x^2 - 5x - 3 \) is also a polynomial function, which is differentiable everywhere. Therefore, it is differentiable on the open interval \((-1, 3)\). 3. Let's calculate \( g(-1) \) and \( g(3) \) to check if they are equal: - \( g(-1) = (-1)^3 - (-1)^2 - 5(-1) - 3 = -1 - 1 + 5 - 3 = 0 \) - \( g(3) = 3^3 - 3^2 - 5(3) - 3 = 27 - 9 - 15 - 3 = 0 \) Since \( g(-1) = g(3) = 0 \), the function \( g(x) \) satisfies the conditions of Rolle's Theorem on the interval \([-1, 3]\). Therefore, Rolle's Theorem applies to the function \( g(x) = x^3 - x^2 - 5x - 3 \) on the interval \([-1, 3]\), and the point guaranteed to exist by Rolle's Theorem is \( x = 0 \). So, the correct choice is: A. Rolle's Theorem applies and the point guaranteed to exist is \( x = 0 \).