CK12 Calculus - Single Variable

Let’s observethat the converseof the last theoremis not necessarilytrue: If we consider and
its graph,then we see that while at is not an extremepointof the function.

Rolle’s Theorem: If is continuousand differentiableon a closedinterval and if
then has at leastone value in the openinterval suchthat.
The proofof Rolle'sTheoremcan be foundat http://en.wikipedia.org/wiki/Rolle's_theorem.

MeanValue Theorem: If is a continuousfunctionon a closedinterval and if containsthe open
interval in its domain, then there exists a number in the interval such that

Proof: Considerthe graphof and secantline as indicatedin the figure.

By the Point-Slopeform of line we have

and

For each in the interval let be the verticaldistancefrom line to the graphof Thenwe
have

for every in