The first derivative g of a twice–differentiable function f is defined for −6≤x≤7. On that interval the graph of g consists of three straight-line segments that join the points (−6,4), (−2,0), (3,2), and (7,−1). Between consecutive points, g is linear with no other features. Assume g is continuous on [−6,7] and differentiable on each open subinterval determined by the listed x–coordinates.
(1) Determine all intervals on [−6,7] where the function f is increasing and all intervals where f is decreasing. State each interval using proper interval notation and justify your answer by referencing the sign of g. No sketch is required.
(2) Identify the x–coordinates of all relative extrema of f on (−6,7). For each extremum, classify it as a relative maximum or a relative minimum and provide a justification based on the behavior of g in a neighborhood of the point.