Mean Value Theorem

 

Motivation:

                        Mean Value Theorem ( MVT ) is used to prove other important theorems. Basically it says : given an secant line of a continuous and differentiable curve, we can find a tangent line with the same slope of the secant line and crossing the curve at the point between the two crossing points of the secant line.

 

                                         

 

                         Intuitively,  it is quite straightforward. But there are a few problems when you try to think more about that ?  First, what does the curve of  “differentiable” function look like?  Second,  there are so many kind of curves that might not even be expressed in any form.  To prove a theorem really needs a lot of imagination ( for generations ).  For completeness, we just prove the theorem very quickly and move to other theorems.

 

 

Lemma ( Rolle’s Theorem )

           Let  f  be a continuous  function on [a.b] and differentiable on (a,b)  such that   f(a)=f(b)=0 .  Then there exists a point a point c in (a,b) such that  f’(c) = 0 .

 

        Proof:

 

                          Since  f  is a continuous function on a closed interval [a.b],  the maximum and minimum must exist in the interval.  So, we can try to consider the following cases.

 

              Case I:

 

                          f(a) = f(b) = 0 .  Thus,  if maximum and minimum happen at the end point at the same time,  it means  f(x) is virtually a constant function on the interval [a,b] , i.e., f(x)=0 in [a,b].  In this case, definitely there exists c such that  f’(c) = 0 .

 

              Case II:

                         For the other situations, it means the maximum or minimum will happen in (a,b) .   Since  f  is differentiable,  either the point of  maximum or minimum will make the derivative equal to 0 at that point.  In other words,  there exists  c in (a.b) such that

 

                                                             f’(c) = 0  .

 

 

 

 

Theorem ( Mean Value Theorem )

                Let  f(x)  be a continuous function on a closed interval  [a,b ]  and differentiable in (a.b) .  Then  there exists c  in (a,b) such that

 

                                                     =

 

                                  

 

        Proof:

                           Let     I(x)=  f(x) – ( f(a) +  (x-a) ) .

 

                            Please note that  I(x) is just the difference between f(x) and the line equation of L in the diagram above.    Thus  I(x) is also a continuous function on [a.b] and differentiable in (a,b) .

 

                           Furthermore,  I(a) = I(b) = 0 .  From Rolle’s Theorem above, we have

 

                               I’(c) = 0 where c is in (a,b) .

 

                        And 

                                     I’(x) = f’(x) -

 

                                         f’(c) =