Endpoints of Intervals Where a Function is Increasing or Decreasing

Date: 05/01/2009 at 23:36:06
From: Bizhan
Subject: No equation or formula applies.

Why do some calculus books include the ends when determining the
intervals in which the graph of a function increases or decreases
while others do not?  I feel that the intervals should be open and the
ends should not be included as they may be, for example, stationary
points where a horizontal tangent can be drawn.  

I noticed that the AP Central always include the ends in the formal
solutions of such problems (one can see many examples there), but an
author like Howard Anton never does.  Could you clarify this for me
please?  Can both versions be correct?

Date: 05/02/2009 at 01:07:07
From: Doctor Minter
Subject: Re: No equation or formula applies.

Hello Bizhan,

You pose an excellent question, and I agree that the discrepancy 
among the various textbooks is quite misleading.

I completely agree with your claim that the intervals should be open 
(that is, should not include the endpoints).  Let me attempt to give 
comprehensive reasoning as to why this should be.

We use derivatives to decide whether a function is increasing and/or 
decreasing on a given interval.  Intervals where the derivative is 
positive suggest that the function is increasing on that interval, 
and intervals where the derivative is negative suggest that the 
function is decreasing on that interval.

Recall that the definition of the derivative f'(x) of a function f(x) 
is given by

                f(x+h) - f(x)
  f'(x) = lim   -------------.
          h->0        h

For the function to be differentiable, this limit MUST EXIST.  The 
concept of the existence of this limit has some very major 
implications.

To find a derivative using the limit definition (which ALL derivatives
must satisfy--the familiar, simpler rules of differentiation such as 
the power rule can all be derived from this definition), we must 
consider what happens as "h" approaches zero from BOTH SIDES.  Recall 
that for a limit to exist, the expression must be the same when the 
value is approached from either side.

Thus, in this case,

                f(x+h) - f(x)         f(x+h) - f(x)
  f'(x) = lim   ------------- = lim   -------------. 
          h->0+       h         h->0-       h  

where the first limit involves values of "h" that are positive, and 
the second limit involves values of "h" that are negative.  These 
one-sided limits must be equal for the derivative to exist.

Thus, for intervals where the function is increasing, it must be true 
that the function is increasing around some small neighborhood that 
encloses each point in the interval.

However, let's consider the right endpoint of a given interval on 
which the function f(x) is increasing.  At values greater than this 
endpoint, one of two things occurs.  Either the function ceases to 
increase, and (like you mentioned) there is a horizontal tangent at 
this point (which is then called a "critical point"), or there is a 
discontinuity in the derivative function f'(x), arising from a 
discontinuity in the function itself, or a "cusp," in which there is 
an abrupt change in slope.  For a discontinuity, the derivative does 
not exist at that point.  Something that does not exist cannot be 
labeled as positive, so the function cannot be said to be increasing 
at this point. 

Either way, critical point or discontinuity, we CANNOT define a 
neighborhood around the endpoint in which the derivative is strictly 
positive!  Thus, it is not safe to include endpoints on intervals of 
increasing or decreasing functions.

In summary, for a function to be increasing (all of these concepts 
are similar for decreasing intervals as well), we have to be able to 
show that the function is greater for larger values of "x," and less 
for smaller values of "x" in a small neighborhood around each point 
in the interval.  An endpoint cannot have both of these properties.

I hope this helps.  Please feel free to write again if you need 
further assistance, or if you have any other questions.  Thanks for 
using Dr. Math!

- Doctor Minter, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/02/2009 at 02:30:03
From: Bizhan
Subject: Thank you (No equation or formula applies.)

Dear Dr. Minter,
 
Your explanations were very clear and helpful.  I enjoyed reading
them.  Thank you for helping me in such a thorough manner.