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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.



Date: 10/22/2014 at 13:51:50
From: Kevin
Subject: Endpoints of intervals for increasing/decreasing (redux)

Dear Dr. Math, 

Above, you gave an argument for why endpoints should *not* be included
when determining intervals where a function is increasing or decreasing.

Implicit in your answer is that "increasing at a point" means "has a
positive derivative in a neighborhood of that point." I wonder if it makes
sense to define increasing at a point. I also wonder about another
definition that I came up with (which, I acknowledge, doesn't work for a
point).

We could define increasing for an interval [a, b] as: whenever x and y are
in [a, b] then f(x) < f(y).

This makes no reference to derivatives, so you could still talk about a
function being increasing even if it fails to be differentiable some
places (e.g., we could say x^(1/3) is increasing everywhere). I'm also
thinking about piecewise functions with jumps; again, it seems like we
should be able to say they're increasing even if the derivative doesn't
exist everywhere.

With this second definition, it seems to me that if a function is
increasing on (a, b) and continuous at a and b, then it would be
guaranteed to be increasing on [a, b].

What do you think? Do we need to have a definition of "increasing at a
point" for some reason? Is there any way to reconcile these two
definitions?

There doesn't seem to be consensus here. It's a basic calculus concept and
there seems to be two (very convincing) ways of looking at it that are in
conflict.



Date: 10/22/2014 at 17:04:02
From: Doctor Peterson
Subject: Re: Endpoints of intervals for increasing/decreasing (redux)

Hi, Kevin.

Dr. Minter is talking within the realm of calculus, which looks at
individual points. As you mention below, this is not really sufficient for
the general case.

> We could define increasing for an interval [a, b] as: whenever x and y
> are in [a, b] then f(x) < f(y).
> 
> This makes no reference to derivatives, so you could still talk about a
> function being increasing even if it fails to be differentiable some
> places (e.g., we could say x^(1/3) is increasing everywhere). I'm also
> thinking about piecewise functions with jumps; again, it seems like we
> should be able to say they're increasing even if the derivative doesn't
> exist everywhere.

This is the proper definition of increasing on an interval, which applies
to any function, and is found, for example, here:

    http://mathworld.wolfram.com/IncreasingFunction.html

    A function f(x) increases on an interval I if f(b) ≥ f(a) for all
    b > a, where a, b in I. If f(b) > f(a) for all b > a, the function is
    said to be strictly increasing.

    ...

    If the derivative f'(x) of a continuous function f(x) satisfies
    f'(x) > 0 on an open interval (a, b), then f(x) is increasing on
    (a, b). However, a function may increase on an interval without
    having a derivative defined at all points. For example, the
    function x^(1/3) is increasing everywhere, including the origin
    x = 0, despite the fact that the derivative is not defined at that
    point. 

> With this second definition, it seems to me that if a function is
> increasing on (a, b) and continuous at a and b, then it would be
> guaranteed to be increasing on [a, b].
> 
> What do you think? Do we need to have a definition of "increasing at a
> point" for some reason? Is there any way to reconcile these two
> definitions?

I agree that these are two different things. The last paragraph From
MathWorld above reflects the relationship between them.

> There doesn't seem to be consensus here. It's a basic calculus concept
> and there seems to be two (very convincing) ways of looking at it that
> are in conflict.

There are actually two different concepts: a precalculus concept,
applicable to any function; and a calculus concept, applicable to
differentiable functions. I would prefer not to confuse them. Much as we
distinguish uniform vs. pointwise continuity, these notions of increasing
could be better distinguished like this:

   The function f is increasing on the interval [0, 1], meaning that
   comparing any two points, the one on the right is higher.

   The function f is increasing at every point on the interval (0, 1),
   meaning that the derivative is positive everywhere.
  
Your question reminds me of an exchange I had last year about a similar 
issue. A student wrote in:

      I have y = f(x). On the x-axis there are points a, b, and c.
      When x = a, y = 0; when x = b, y = 4; when x = c, y = 1.
      
      I realise the function increases on [a, b].
      I also realise the function decreases on [b, c].
      
      But why is the b in brackets? I know that they indicate closed 
      intervals; that's no problem. If the graph increases from point a 
      to point b, that is [a, b], but then the graph *MUST* decrease 
      on (b, c].
      
      If it increases TO "b," it decreases FROM "b" ... EXCEPT "b"?

I responded:

      I'll suppose you know that a < b < c, so the picture is something
      like this:
      
         +            o
         |
         +
         |
         +
         |
         +                   o
         |
         +-----o------+------+-----
               a      b      c
       
      We can't be sure of this just from only the facts you've given me;
      I'll suppose you were told this explicitly, or that you were given a
      graph that makes it clear, like this:
      
         +            o
         |
         +        o        o
         |
         +       o          o
         |
         +      o            o
         |
         +-----o------+------+-----
               a      b      c
      
      I think what you are asking is why they include the endpoints in the
      intervals. That's strange, because we've had other questions about
      why the endpoints are NEVER included in the interval of increase or
      decrease!
      
      Different texts have different policies on this. How does your text
      DEFINE "increasing on an interval"? Can you show me the first
      example they give?
      
      See these pages, which emphasize this variability among texts:
      
        Brackets or Parentheses?
          http://mathforum.org/library/drmath/view/53566.html
      
        Endpoints of Intervals Where a Function is Increasing 
        or Decreasing
          http://mathforum.org/library/drmath/view/73202.html
      
      I'm inclined to agree more with the first of these than the second;
      but I think in pre-calculus, it's a good idea to ignore this detail
      and either always use open intervals or always use closed intervals.
      It's not really an important issue; but your concern that a function
      can't be increasing AND decreasing at the same point would tilt me
      in the direction of using open intervals just to avoid confusing
      students like you!
      
      Really, however, you need to notice that your definition probably is
      only about increasing or decreasing IN AN INTERVAL, not AT A POINT.
      That is, they are not saying the function is increasing at b -- only
      that it is increasing in the interval [a, b]. So no claim is being
      made that the function is both increasing and decreasing at b!
      
      Once I see your book's definition, I can be more clear on that.

This student never replied with his text's definition, so I didn't get to
explore the details with him. One such detail would have been the
distinction between saying that a function is increasing on an interval
and saying that that is a maximal interval, in the sense that there is no
containing interval (open or closed) on which it is increasing.

In my experience, texts leave a lot unstated. While these omitted details
might keep things simple for the less mature student, they would be worth
exploring with a curious, capable one like you!

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



Date: 10/22/2014 at 17:13:02
From: Kevin
Subject: Thank you (Endpoints of intervals for increasing/decreasing (redux))

Thanks. The link you posted seemed to confirm what I thought was right.

And you pinpointed the issue: it usually arises from vague definitions in
texts.

Thanks for your time!
Associated Topics:
College Calculus

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