A continuous function is differentiable on a closed interval
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A continuous function is differentiable on a closed interval [a,b]. Mathematicians often claim that to locate the function’s absolute extreme values on the interval, one need only consider its critical points (where the derivative is zero or undefined) together with the endpoints.
Answer the following questions to discuss the logical foundation of that claim.
(a) State the Extreme Value Theorem in your own words, clearly indicating the hypotheses and the conclusion.
(b) Give a concise argument, rooted in the definition of derivative, explaining why a point c in (a,b) at which the function attains a local maximum or minimum must satisfy f′(c)=0.
(c) Explain why evaluating the function solely at the interior critical points and at x=a and x=b is sufficient to identify every absolute (global) maximum and minimum of the function on [a,b]. Your explanation should connect parts (a) and (b).