- , in the interval
- Statewhetherthe indicatedx-valuescorrespondto maximumor minimumvaluesof the functiondepicted
below. - Provethe IntermediateValue Theorem:If a functionis continuouson a closedinterval[a, b], then the
functionassumeseveryvaluebetweenf(a) andf(b).
Answers
- Whilegraphof the functionappearsto be continuouseverywhere,a checkof the tablevaluesindicates
that the functionis not continuousatx= -1. - Whilethe functionappearsto be continuousfor allx= -2, a checkof the tablevaluesindicatesthat the
functionis not continuousatx= 2.
3.
doesnot exist
By the IntermediateValue Theorem,thereis anx-valuecwith
f(c) = 0.
By the IntermediateValue Theorem,thereis an x-valuec with f(c)
= 0.
9.x = ais a relativemaximum,x = bis an absoluteminimum,x=cis an absolutemaximumandx=dis
not a maximumnor a minimum.
Hereis an outlineof the proof:we needto showthat for everynumberdbetweenf(a) andf(b), there
existsa numbersuchthatf(c) =d. 1) Assumethatf(a) <f(c) <f(b). 2) LetSbe the set ofxε [a, b]> for which
f(x) <d. Notethataε S,bε S. sobis an upperboundfor setS. Henceby the completenesspropertyof the