All questions

70. For what values of c does the polynomial P(x) = x* + ex' + x have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases? 71. Prove that if (c, f (c)) is a point of inflection of the graph of f and f" exists in an open interval that contains c, then f"(c) = 0. [Hint: Apply the First Derivative Test and Fermat's Theorem to the function g = f'.] 72. Show that if f(x) = x4, then f"(0) = 0, but (0, 0) is not an inflection point of the graph of f. 73. Show that the function g(x) = x | x | has an inflection point at (0, 0) but g"(0) does not exist. 74. Suppose that f" is continuous and f'(c) = f"(c) = 0, but f"(c) > 0. Does f have a local maximum or minimum at c? Does f have a point of inflection at c? 75. Suppose f is differentiable on an interval I and f'(x) > 0 for all numbers x in I except for a single number c. Prove that f is increasing on the entire interval