Optimization problems often require analyzing critical points and endpoints of a continuous single-variable function defined on a closed interval. Consider such a function f defined on [a,b], where f arose from applying a real-world constraint to an initial multivariable situation.
(a) Explain why both interior critical points of f and the endpoints x=a and x=b must be examined when determining the absolute maximum or minimum of f on [a,b].
(b) Describe how the first derivative and second derivative provide different information about the behavior of f near a critical point, and discuss why neither test alone is always sufficient for identifying absolute extrema.
(c) Give one concrete example (without fully solving it) of a physical or geometric scenario where failing to check an endpoint would lead to an incorrect conclusion about the optimal solution, and briefly explain the oversight.