The function f is represented graphically by three straight-line segments on the closed interval 0≤x≤6.
• From (0,0) to (2,4) the graph is a line segment of positive slope.
• From (2,4) to (4,4) the graph is horizontal.
• From (4,4) to (6,0) the graph is a line segment of negative slope.
Throughout, the endpoints shown are included in the graph, and no other points are plotted.
Answer the following, using appropriate derivative notation and complete justification.
(a) Calculate the average rate of change of f on each of the intervals [0,2], [2,4], and [4,6]. Express each answer as a simplified constant.
(b) For each open sub-interval (0,2), (2,4), and (4,6), determine the exact value of the instantaneous rate of change f′(x). Write a single piecewise definition for f′(x) on 0<x<6.
(c) For x=2 and x=4, decide whether f′(x) exists. Provide a clear calculus-based justification for each endpoint.