A function h is twice differentiable for all real numbers. A portion of the graph of h′′ is described as follows:
• For −3≤x≤4, the graph of h′′ consists of the top half of a circle of radius 5 centered at (0,−1).
Beyond that interval, h′′ is known only through the following statements.
• For x>4, h′′(x)>0 and h′′(x)→0 as x→∞.
• For x<−3, h′′(x)<0 and h′′(x)→0 as x→−∞.
Answer the questions below about the concavity of h.
(a) Determine the concavity of h on the interval −3<x<4. Justify your answer by referring to the sign of h′′ on that interval.
(b) Combine the information given for x<−3 and x>4 with your result from part (a) to state the interval(s) of concave up and concave down behavior for h on the entire real line.
(c) Identify all x–values at which the graph of h has points of inflection. Give a reason for each answer.
(d) Discuss how the concavity information from parts (a)–(c) constrains the overall shape of the graph of h. In your discussion, comment on the behavior of h for very large positive and very large negative values of x.