The twice-differentiable function g is defined for all real numbers. Its second derivative g′′ is known only through the following description:
• g′′(x)>0 for x<−3
• g′′(x)<0 for −3<x<1
• g′′(x)>0 for 1<x<4
• g′′(x)<0 for x>4
Answer the following questions about the graph of g.
(a) Describe the concavity of the graph of g on each of the four intervals determined by x=−3,1,4. Give reasons for each conclusion.
(b) Identify all real numbers x at which the graph of g could have a point of inflection. Explain your reasoning.
(c) For each candidate from part (b), state whether a point of inflection actually occurs and justify your decision in terms of changes in concavity.