MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A differentiable function p is defined for −4<x<6. The graph of p′ is described below.
• On −4<x<−1, p′ decreases linearly from 6 to 0.
• At x=−1, p′ has a sharp corner (cusp).
• On −1<x<3, p′ increases smoothly from 0 to 8.
• At x=3, p′ has a vertical tangent but remains continuous.
• On 3<x<6, p′ decreases smoothly from 8 to −4.
No other critical behavior occurs for p′ in (−4,6).
(a) Determine each open interval in (−4,6) on which p is concave up and each interval on which it is concave down.
(b) Identify all x-values in (−4,6) at which p has a point of inflection, and classify whether the point occurs where p′ is differentiable or where p′ fails to be differentiable.
(c) The behavior of p′ at x=−1 and x=3 is non-smooth. Explain how concavity of p can still be determined at such points, and justify whether either of these points can be an inflection point of p.