Let be the first derivative of a twice–differentiable function
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let g be the first derivative of a twice–differentiable function f. The behavior of g is summarized verbally below.
• g is strictly decreasing on (−∞,−2).
• g is strictly increasing on (−2,1).
• g is strictly decreasing on (1,4).
• g is strictly increasing on (4,∞).
Assume g is continuous and differentiable everywhere and that its local extrema occur only at x=−2,1,4.
(a) Identify the x–coordinates of all critical points of g. Classify each as a local maximum or a local minimum of g, and explain your reasoning.
(b) Using only the information given, list all x–values that are candidates for inflection points of f. Justify why these x–values merit consideration.
(c) For each candidate identified in part (b), determine whether f actually has an inflection point there. Support each conclusion with a clear concavity argument based on the behavior of g. State the intervals on which f is concave up and concave down.