MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A twice-differentiable function f is defined for −4≤x≤6. A selected set of values of its first derivative f′(x) is shown in the table below.
x
-4
-2
0
2
4
6
f'(x)
-7
-3
0
2
5
7
Assume that f′(x) is continuous on the given interval and changes smoothly between the tabulated points.
(a) On which subintervals of [−4,6] is f concave up and on which subintervals is f concave down? Indicate your answers using interval notation.
(b) Describe the reasoning that connects the behavior of f′(x) in the table to the concavity of f. Your explanation should reference the appropriate derivative and its sign.
(c) Identify all x-coordinates in [−4,6] at which f has a point of inflection, and justify each selection.