Let g(x)=∣x2−4x∣ on the closed interval [−1,5].
(a) Rewrite g as an explicit piecewise–defined function without the absolute value symbol, clearly indicating the interval on which each branch is valid.
(b) Determine every point in (−1,5) where g fails to be differentiable. Show all reasoning, including derivative calculations or limits from each side as appropriate.
(c) Decide whether the hypotheses of the Mean Value Theorem are met on [−1,5]. Provide a logical argument that references your work in parts (a) and (b).
(d) If you concluded in part (c) that the hypotheses are satisfied on any subinterval of [−1,5], select one such subinterval and find at least one value c in the interior of that subinterval for which g′(c) equals the average rate of change of g over that subinterval. Show all supporting calculations.