The derivative g=f′ of a twice-differentiable function f is represented by straight-line segments connecting the points (−4,0), (−2,4), (0,0), (2,−2), and (4,0).
Assume that g is linear between consecutive points listed above and that no other critical points or discontinuities occur. The value of the original function at x=−4 is f(−4)=0.
(a) Construct a complete sign chart for g on the interval [−4,4].
(b) Using your sign chart, identify each critical number of f in (−4,4). Classify every critical point of f as a relative maximum or relative minimum and justify your classifications.
(c) Determine the intervals on which f is concave up and the intervals on which it is concave down. Indicate every x-coordinate of a point of inflection and support your conclusions.
(d) On the axes you provide, sketch a possible graph of f on the interval [−4,4] that is consistent with all information obtained in parts (a)–(c). Clearly label each relative extremum and point of inflection.