MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
For a differentiable function h, selected values of its first derivative h′(x) are given in the table below.
x
0
1
2
3
4
5
h′(x)
−3
−1
1
2
1
−2
Assume that h′(x) is continuous and strictly increasing on [0,3] and strictly decreasing on [3,5].
(a) Using the data in the table, indicate the open intervals on 0<x<5 where h is increasing. Explain how the signs of the tabulated values support your answer.
(b) Without calculating an explicit formula for h′′(x), determine the open intervals on 0<x<5 where h is concave down. Base your reasoning on the stated behavior of h′(x).
(c) From your answers to parts (a) and (b), state the open intervals on 0<x<5 where h is both increasing and concave down. Provide complete justification in terms of h′ and its variation.