Consider the function
h(x)=∣x2−4∣,−∞<x<∞.
The function h is continuous everywhere but has points at which its derivative does not exist.
(a) Express h(x) as an explicit piecewise-defined function without the absolute value symbol. Clearly indicate the domain of each branch.
(b) Compute h′(x) wherever it exists. Identify all x-values in (−∞,∞) at which h′ is undefined.
(c) List all critical numbers of h and create a sign chart for h′ on the real line, marking the critical numbers and points where h′ fails to exist.
(d) Using your sign chart, determine the intervals on which h is increasing and the intervals on which it is decreasing.
(e) Classify each critical number as yielding a relative maximum, a relative minimum, or neither. Provide justification based on your analysis of h′.