9.
- a.
b.
Extremaand the MeanValue Theorem
LearningObjectives
A studentwill be able to:
- Solveproblemsthat involveextrema.
- StudyRolle’s Theorem.
- Use the MeanValue Theoremto solveproblems.
Introduction
In this lessonwe will discussa secondapplicationof derivatives,as a meansto studyextreme(maximum
and minimum)valuesof functions.We will learnhow the maximumand minimumvaluesof functionsrelate
to derivatives.
Let’s start our discussionwith someformalworkingdefinitionsof the maximumand minimumvaluesof a
function.
Definition:A function has amaximumat if for all in the domainof Similarly,
has aminimum at if for all in the domainof The valuesof the functionfor
these -valuesare called extreme valuesorextrema.
Hereis an exampleof a functionthat has a maximumat and a minimumat :
Observethe graphat. Whilewe do not havea minimumat , we note that for
all near We say that the functionhas alocalminimumat Similarly, we say that the function
has a localmaximumat since for some containedin openintervalsof
Let’s recallthe Min-MaxTheoremthat we discussedin lessonon Continuity.