Let be a three–times differentiable function whose graphs of and are described below
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let h be a three–times differentiable function whose graphs of h and h′′ are described below. No direct information about h′ is provided.
Graph of h (verbal description):
• h has x–intercepts at x=−4,0,4.
• h has relative maxima at x=−2 and x=2 and a relative minimum at x=0.
• h is concave up on (−4,−1) and (1,4) and concave down on (−1,1), with inflection points at x=−1 and x=1.
Graph of h′′ (verbal description):
• h′′ is positive on (−3,−1) and (1,3), negative elsewhere on (−4,4).
• h′′ crosses the x–axis at x=−3,−1,1,3.
• h′′ attains a relative maximum at x=−2 and a relative minimum at x=2.
(a) Construct a carefully labeled sign chart for h′ on the interval (−4,4) using only the information provided. Show all reasoning.
(b) Using your sign chart from part (a), determine all intervals where h is increasing and all intervals where it is decreasing. Explain how the patterns in h′′ reinforce your conclusions.
(c) The graph of h′ is not given. Describe all qualitative features (zeros, relative extrema, concavity) that the graph of h′ must exhibit on (−4,4). Provide an analytical justification for each stated feature.