A common claim in introductory calculus texts is: "If a function is continuous on
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A common claim in introductory calculus texts is: "If a function is continuous on [a,b], differentiable on (a,b), and has exactly one critical point c∈(a,b) at which f′′(c)<0, then f(c) is the absolute maximum value of f on [a,b]." Investigate the robustness of this claim by analyzing the necessity of each hypothesis.
(a) Explain in general terms why the conclusion need not hold if the function is not continuous at an endpoint of the interval. Illustrate your explanation with a sketch or verbal description of a possible graph.
(b) Construct an explicit function g on a closed interval [p,q] that is continuous on [p,q] and differentiable on (p,q), has more than one critical point in (p,q), and has g′′(d)<0 at one of those critical points d, yet fails to attain an absolute maximum at d. Provide full justification for each property of your example.
(c) Suppose instead that h is continuous on [u,v], differentiable on (u,v), has exactly one critical point e in (u,v) with h′′(e)<0, but the interval [u,v] is unbounded above or below. Discuss whether h(e) must still be the largest value of h on that interval and support your answer with a formal argument or counterexample.
(d) Summarize which hypotheses are indispensable for the original claim and provide a concise proof of the claim under the minimal set of hypotheses you have identified.