Answer all the 8 questions in the photo

AP 1. Multiple Choice Which of these is a necessary condition for applying the Mean Value Theorem to a function f on an interval [a7 17]? I. f is continuous on [cab]. nan) =f(b)- 111. f has either a local minimum or a local maximum on [a. b]. (A) I only (B) l and 11 only (C) II and Ill (D) I. TI, and Ill 7. Let x) : v4.1: + 1. What is the value ofc in the interval [0. 6] whose existence is guaranteed by the Mean Value Theorem? (A) l (B) 2 (C) 3 (D) 4 1720 Verify that the function satises the hypotheses of the Mean Value Theorem on the given interval. Find all numbers c that satisfy the conclusion of the Mean Value Theorem. 17. f(x) = 2x2 , 3x +1, [0, 2] 23. Let x) : (x , 3)'2. Show that there is no value c in (1,4) such that f(4) f(l) :f"(c)(4 1) Explain why this does not contradict the Mean Value Theorem. AP29. A particle moves along a horizontal line so that its posi- tion at time t, t 2 0, is given by i'(t):t'1 9):2 -l 24t 10, where I is measured in seconds and s in meters. (3) Find the average velocity of the particle over the inter val [0, 2]. Does the particle travel with this velocity at any instant on the interval? Explain the reason for your answer. (b) Write an expression for '00), the velocity of the particle at time t. (c) Find the values of t for which the velocity is zero. (d) Using your answer from part (c), explain why there must he a time I at which the acceleration is zero. Find this value. 35. If f(1) = 10 and f'(x) 2 2 for l g x g 4, What is the smallest possible value of f(4)? 37. Can you nd afunetion f such that f(0) : 1, f(2) : 4, and f'(x) g 2 for all x? Explain your reasoning. 46. At 2:00 PM a car's speedometer reads 30 mi/h. At 2le PM it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/hz