The table below lists selected values of the derivative for a twice-differentiable…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The table below lists selected values of the derivative f′(x) for a twice-differentiable function f defined on the interval −4≤x≤6. At the listed x–values, f′(x) is either positive (+), negative (−), or undefined (DNE). Between the listed points, f′(x) does not change sign except as indicated.
x
−4
−2
0
1
3
5
f′(x)
+
0
−
DNE
+
0
Beyond x=5, the derivative remains negative until x=6.
(a) Describe all intervals on which the function f is increasing and all intervals on which it is decreasing. Explain your reasoning using the information in the table.
(b) Identify each x-value in the table at which f attains a relative maximum or a relative minimum. For each extremum, classify it as a maximum or a minimum and justify your conclusion.
(c) At x=1 the derivative is undefined. Explain what type of critical point this represents for f and determine whether a relative extremum occurs there. Support your answer with appropriate reasoning.