CK12 Calculus - Single Variable

Notethat Since is continuousin and existsin then Rolle’s Theorem
applies.Hencethereexists in with

So for every in
In particular,

and

The proofis complete.
Example2:

Verify that the MeanValue Theoremappliesfor the function on the interval

Solution:

We needto find c in the interval suchthat

Notethat and Hence,we mustsolvethe following
equation:

By substitution,we have