Mean value theorem, extreme value theorem and critical points

Learning objectives

State the mean value theorem, including its hypotheses of continuity on [a, b] and differentiability on (a, b), and interpret it graphically as a tangent parallel to a secant

Apply the mean value theorem to justify that a derivative takes a particular value somewhere in an interval, for example from a table of values or in a motion context, and recognize Rolle's theorem as the special case f(a) = f(b)

Explain why the conclusion of the mean value theorem may fail when a hypothesis fails, and check the hypotheses explicitly before applying the theorem

State the extreme value theorem and explain why a continuous function on a closed interval must attain both an absolute maximum and an absolute minimum

Define a critical point as a point in the domain where f′ is zero or undefined, find all critical points of a function, and distinguish between local (relative) and global (absolute) extrema