Mean value theorem, critical points and the first derivative test

Learning objectives

State the hypotheses and conclusion of the mean value theorem, namely continuity on the closed interval and differentiability on the open interval, and interpret it as a tangent parallel to the secant

Apply the mean value theorem to guarantee that a derivative takes a given value somewhere in an interval, and recognize Rolle's theorem as the special case in which the endpoint values are equal

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

Identify critical points as interior points where the derivative is zero or undefined, build a sign chart for f′, and state the intervals on which f increases or decreases

Apply the first derivative test to classify each critical point, and use the candidates test, comparing values at critical points and endpoints, to find absolute extrema on a closed interval

Deduce the behavior of f from a graph of f′ alone, and write justifications that cite the sign change of the derivative rather than the shape of a picture