1. The problem statement, all variables and given/known data Suppose that a continuous function f(x) has horizontal tangent lines at x = -1, x = 0, and x = 1. If f"(x) = 60x^3 - 30x, then which of the following statements is/are true? A) f(x) has a local max at x = 1 B) f(x) has a local min at x = -1 C) f(x) has an inflection point at x = 0 2. Relevant equations Local maximums occur at critical points. All points at which horizontal tangent lines occur are critical points because the existence of a horizontal tangent line at that point implies the existence of that point on the function, and as we know, critical points must exist in the domain of the function. Therefore, x = 1, 0, and 1 are critical points. We can use the second derivative test to test for local extrema. 3. The attempt at a solution f"(-1) = - 30. x = -1 is a local max. B is true. f"(1) = 30. x = 1 is a local min. A is true. f(x) has an inflection point at x = 0; the second derivative is 0 at x = 0 and x = ±1/sqrt(2). f"(x) changes sign around x = 0 from being positive in the interval (-1/sqrt(2), 0) and (0, 1/sqrt(2)). Therefore, all three are true.