Let h be a differentiable function whose derivative h′ is described by the following qualitative information:
• h′(x)=0 at x=−2,1, and 5.
• h′(x)>0 on the intervals (−∞,−2),(1,5), and (5,∞).
• h′(x)<0 on the interval (−2,1).
• The values of h′ are increasing on (−∞,−2), decreasing on (−2,1), increasing on (1,5), and decreasing on (5,∞).
Assume h′′ exists wherever needed.
(a) Determine the open intervals on which h is increasing and on which it is decreasing. Justify your answer based on the sign of h′.
(b) Identify all critical points of h and classify each as a local maximum, local minimum, or neither. Support your conclusion with appropriate reasoning.
(c) Using the behavior of h′ given above, determine the open intervals on which h is concave up and on which it is concave down. Explain how you arrived at your answer.
(d) Sketch one possible graph of h that is consistent with all of the information provided. Clearly label the critical points, points of inflection, and indicate relative scales on the axes.