Suppose 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
Suppose a function f is continuous on the closed interval [a,b] and differentiable on (a,b). Assume that f has exactly one critical point c in (a,b).
(1) Part 1: (a) Explain why, if f′′(c)>0, then f(c) must be the absolute minimum value of f on [a,b]. Your explanation should reference any relevant existence principles for extrema on closed intervals.
(2) Part 2: (b) Describe how the conclusion changes when f′′(c)<0, providing reasoning analogous to part (a).
(3) Part 3: (c) Give a brief argument or explicit example to show that if the hypothesis of continuity on [a,b] is removed, the conclusion in part (a) can fail even when f′′(c)>0.