>I'm a newbie to AP Calculus, stepped into the class mid-semester due to a teacher resigning. I'm grading a quiz from the old teacher, and it has the following problem: > >A function f is continuous on the closed interval [2,4] and twice differentiable on the open interval (2,4). If f'(3)=2 and f"(x)<0 on the open interval (2,4), which of the following could be a table of values for f? > >A) (2,2.5),(3,5),(4,6.5) B) (2,2.5),(3,5),(4,7) C) (2,3),(3,5),(4,6.5) D) (2,3),(3,5),(4,7) > > > > On approach is the graph connected scatter plots of each of the four data sets. Each plot will be of piece wise linear functions that approximate the graph of f. Then sketch the line tangent to the graph of f at (3, f(3)) and note that since, the second derivative has negative values throughout the open interval, the tangent line slopes are decreasing. Then compare the slopes of each of the two linear segments in each plot. The slopes of these segments should be decreasing from left to right, just as the tangent line slopes to the graph of f are decreasing. Furthermore, the slope of the right most segment must be less than 2. You could support this further by applying the Mean Value Theorem on the sub intervals [2,3] and [3,4]. Only choice A is consistent with the conditions stated in this excellent question.