Rolle's Theorem is valid for the given function f(x)=x3+bx2+cx,1≤x≤2 at the point x= 43, then values of b and c are respectively

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) = 3x2 − 6x + 4,   [−1, 3]

Consider the following function.f(x) = 1 − x2/3Find f(−1) and f(1).f(−1)= 0 f(1)= 0 Find all values c in (−1, 1) such that f '(c) = 0. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)c = Based off of this information, what conclusions can be made about Rolle's Theorem?This contradicts Rolle's Theorem, since f is differentiable, f(−1) = f(1), and f '(c) = 0 exists, but c is not in (−1, 1).This does not contradict Rolle's Theorem, since f '(0) = 0, and 0 is in the interval (−1, 1).    This contradicts Rolle's Theorem, since f(−1) = f(1), there should exist a number c in (−1, 1) such that f '(c) = 0.This does not contradict Rolle's Theorem, since f '(0) does not exist, and so f is not differentiable on (−1, 1).Nothing can be concluded

Based off of this information, what conclusions can be made about Rolle's Theorem?This contradicts Rolle's Theorem, since f is differentiable, f(−1) = f(1), and f '(c) = 0 exists, but c is not in (−1, 1).This does not contradict Rolle's Theorem, since f '(0) = 0, and 0 is in the interval (−1, 1).    This contradicts Rolle's Theorem, since f(−1) = f(1), there should exist a number c in (−1, 1) such that f '(c) = 0.This does not contradict Rolle's Theorem, since f '(0) does not exist, and so f is not differentiable on (−1, 1).Nothing can be concluded.

If the function f(x) = ax3 – 6x2 + 11x – 6 satisfies conditions of Rolle’s theorem in [1, 3], then the value of a must be ________.

et f(x) = (x − 3)−2. Find all values of c in (2, 5) such that f(5) − f(2) = f '(c)(5 − 2). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)c = 52​ Based off of this information, what conclusions can be made about the Mean Value Theorem?This contradicts the Mean Value Theorem since f satisfies the hypotheses on the given interval but there does not exist any c on (2, 5) such that f '(c) = f(5) − f(2)5 − 2.This does not contradict the Mean Value Theorem since f is not continuous at x = 3.    This does not contradict the Mean Value Theorem since f is continuous on (2, 5), and there exists a c on (2, 5) such that f '(c) = f(5) − f(2)5 − 2.This contradicts the Mean Value Theorem since there exists a c on (2, 5) such that f '(c) = f(5) − f(2)5 − 2, but f is not continuous at x = 3.Nothing can be concluded.