A function is continuous on the closed interval and differentiable on
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function f is continuous on the closed interval [a,b] and differentiable on (a,b). Students sometimes overlook endpoints when searching for extreme values.
Respond to each of the following prompts to clarify the theoretical foundation for locating absolute extrema on a closed interval.
(a) Define a critical number of a function on an open interval.
(b) Explain, without performing any calculations, why every absolute maximum or minimum value of f on [a,b] must occur either at a critical number in (a,b) or at one of the endpoints a or b.
(c) Construct an explicit continuous function on a finite interval of your choice whose absolute maximum occurs at an endpoint and whose absolute minimum occurs at an interior critical number. State the interval, identify the absolute extrema, and justify your conclusions.