The table below gives selected values of the derivative of a twice-differentiable function
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The table below gives selected values of the derivative p′(x) of a twice-differentiable function p.
x
-4
-3
-2
-1
0
1
2
3
4
p′(x)
5
2
0
-3
-2
0
4
6
7
Assume that p′(x) is continuous between the listed points and that no other zeros or discontinuities occur on −4≤x≤4.
(a) Identify all critical numbers of p in the interval [−4,4] and classify each as coming from p′=0 or from some other condition.
(b) Construct a sign chart for p′ on [−4,4] using the information in the table and the stated assumptions.
(c) Determine every open subinterval of [−4,4] on which p is increasing and every open subinterval on which p is decreasing. Justify your answer.
(d) For each critical number found in part (a), decide whether p has a relative maximum, a relative minimum, or neither at that number. Explain your reasoning.