MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
A function f is differentiable on −6≤x≤8. A verbal description of the graph of f′ is given below.
Graph of f′ (all coordinates are exact):
• f′ is linear on [−6,−2] with f′(−6)=4 and f′(−2)=0.
• f′ is linear on [−2,1] with f′(−2)=0 and f′(1)=−3.
• f′ is linear on [1,4] with f′(1)=−3 and f′(4)=0.
• f′ is constant on [4,6] with f′(x)=0.
• f′ is linear on [6,8] with f′(6)=0 and f′(8)=2.
No other information about f′ is provided; assume the above pieces join exactly at the listed endpoints.
(a) Determine the open intervals on (−6,8) where f′′(x)>0, where f′′(x)<0, and where f′′(x)=0. Justify each conclusion using properties of f′. Your answer should explicitly state every interval and endpoint.
(b) Using your work in part (a), sketch a possible graph of f′′ on the interval −6≤x≤8. Clearly label all key values on both axes, include open or closed dots where appropriate, and indicate any discontinuities.
(c) Identify all x–values in (−6,8) at which f′′ is not defined. For each such value, explain the precise graphical or analytical feature of f′ that causes the non-differentiability of f′′. Be specific in your reasoning.