A differentiable function h has a first derivative h′ whose graph consists of two smooth continuous arcs. The graph of h′ has the following key features:
• h′ increases on (−∞,2), attains a local maximum at (2,5), and decreases on (2,5).
• h′ attains a local minimum at (5,−1) and increases on (5,9).
• h′(x)=0 only at x=2 and x=5.
• h′ is negative for x<0, positive for 0<x<4, negative for 4<x<6, and positive for x>6.
• h′ is continuous and differentiable everywhere.
Answer the following questions about h.
(1) Explain how the given information about h′ can be used to determine the sign of h′′ on the intervals (−∞,2), (2,5), and (5,9).
(2) Using your reasoning from part (a), identify all intervals on which h is concave up and all intervals on which h is concave down.
(3) Determine the x–coordinates of every point of inflection of h. Justify each claim with reference to the behavior of h′.
(4) Sketch a possible graph of h on the interval 0≤x≤9 that is consistent with all given information and your answers to parts (b) and (c). Clearly label each point of inflection on your sketch.