Rolle's theorem Consider the function f(x) given by f(x) x3 - 5x2 + 6x { f(x) = x^3 - 5 x^2 + 6x a) What conclusion does Rolle's theorem give about this function f(x) on the interval [0, 3] ? b) Solve to find all critical point of f(x), as well as all critical points within this interval (0,3). QUESTION 23 mean value theorem Apply the Mean Value Theorem to the function f(x) given below on the interval [0,4). First verify the conditions for the theorem are satisfied. Then state the conclusion of the theroem. f(x) = 6x2 - x3 { f(x) = 6x^2 - x^3 Finally, solve to locate all values of c in the interval (0,4) that satisfy the conclusions of the theorem. QUESTION 24 4 p Mean Value theorem exceptional case where theorem does not apply Consdier the function f(x) given as follows f(x) = 2x-6| { f(x) = .cuberoot( |2x-6| ). %3D Explain why the Mean Value theorem does not apply to f(x) on the interval [-1,7]. Also observe there does not exist any c in the interval (-1,7) such that f(c) = the average rate of change f(7) f(- 1) %3D 7+1 QUESTION 25 increasing / decreasing Consider the function f(x) given by the following f(x) = x4 + 2x3 + x2-5 { f(x) = x^4 + 2x^3 + x^2 - 5.. } %3D a. Find the derivative use this to determine all critical points. b. Determine the regions where f(x) is increasing and regions where decreasing.