For each function f and interval [a, b] in Exercises 56–67, use definite integrals and the Fundamental Theorem of Calculus to find the exact average value of f from x = a to x = b. Then use a graph of f to verify that your answer is reasonable. 56. f() — х — 1, [-1,3] 57. f(x) = 3x + 1, [0, 4] 58. f(x) = 4 – x², [-2, 2] 59. f(x) = 4, [-37.2, 103.75] 60. f(x) = (x + 2)² – 5, [–5,0] 61. f) %3 х2 — 2х - 1, [0,3] 62. f(x) = 4x3/2, [a, b] = [0, 2] 63. f(x) = (e*)², [a, b] = [=1, 1] 1 64. f(x) = [a, b] = [2, 5] Зх + 1' 65. f(x) = 2 – VI, [a, b] = [1,8] 66. f(x) = x² sin(x³ + 1), [a, b] = [-1,2] 67. f(x) = sinx+x cos x, [a, b] = [-1,7] %3D

Let f be a real-valued function on R. Suppose that f(−1) = 3 and f'(x) ≤ 2 for all values ofx. How large can f(5) possibly be?

Determine whether the statement is true or false. There exists a function f such that f(x) > 0, f'(x) > 0, and f"(x) < O for all x. True False