A differentiable function h models the height (in meters) of a drone above the ground as a function of horizontal distance x (in meters) traveled from its launch point on a straight flight path. Engineers recorded the following information from the screen of their flight-analysis software.
Graphical observations (verbal description):
1. For 0≤x≤20, the graph of h is increasing and concave down.
2. At x≈20, the tangent to the graph is horizontal.
3. For 20<x<45, the graph is decreasing; the slope becomes more negative until x≈34, after which the magnitude of the slope decreases.
4. At x≈34, the graph changes concavity from concave down to concave up while continuing to decrease.
5. For 45≤x≤60, the graph is increasing and concave up, and the slope approaches the same magnitude it had near x=10.
No algebraic expression for h is available.
Answer the following questions about the flight profile of the drone.
(a) Identify the x-coordinate(s) of all critical points of h on the interval 0≤x≤60, and explain why each point is a critical point.
(b) Using only the features described above, determine the open intervals on which h is increasing and on which it is decreasing. Justify your answer.
(c) Classify each critical point you identified in part (a) as a local maximum, local minimum, or neither. Provide a justification that references changes in the sign of h′(x) or concavity, as appropriate.
(d) Explain the practical meaning, in the context of the flight, of the local extremum that occurs at x≈20. Include units in your response.