This article I came across while solving a problem which states that is differentiable on and given that . Prove that .

A prof asked his students to solve this problem on which one of his students gave the solution, let . Then



Now this proof is not correct in the sense that an application of the was intended but,in fact, we don’t know that

So this post is on the modification that can be done to student’s proof so as to make it correct.(I call it the mistake which was not mistooken 🙂 )

Consider the three cases in which has a non-zero value(including , has a value of 0, or fails to exists.

In the first case we’ve and L’Hopital’s can be applied as student stated.

If then we cannot use L’Hopital’s Rule because need not have the form . However, by Mean value theorem, we know that there exists,for every positive integer with .

Then . Moreover,since . So L=0.

The last case is a bit tricky, assuming that fails to exists( ), we shall obtain a contradiction. Let denote the set of where has a local extreme value. Note that this set will be infinite and unbounded( because if it is finite then after last extrema function will be monotonic and thus will surely exists(maybe )).

By Rolle’s theorem . Now let be any sequence in with . Since the numbers are not generally in , choose two sequences in tending to and satisfying, for each and either or .

Then from we have It follows that and, hence, . This is a contradiction.

Source:

An Application of L’Hopital’s Rule

Author(s): Jitan Lu

The College Mathematics Journal, Vol. 32, No. 5 (Nov., 2001), pp. 370-372