A student studying extrema on closed intervals makes the following claim:
“Because the function p has a local (relative) maximum at x=2, that point must also be the absolute (global) maximum of p on the interval [0,5].”
(a) Explain why the student’s claim could be incorrect. In your explanation, distinguish clearly between a local maximum and an absolute maximum on a closed interval.
(b) Describe a systematic procedure that guarantees identification of every absolute maximum and absolute minimum of a differentiable function on a closed interval. Your description should reference the role of critical points, the endpoints, and the behavior of the derivative without requiring tabulation of all function values.