



Wherever
there is a "smile" or a "frown,"



there is
usually a local maximum or local minimum.



That is,
a point that has the highest or lowest value in the area. 








These
points are also sometimes called extrema. 


From the
info in the last few chapters we know how to find these points. 


The thing
we don't have is a rule to tell us when these local values 


are the
absolute highest or lowest value the
function will ever have. 





OK,
here's the rule. 


The only
points where the absolute maximum or minimum function values 


can
possible occur are the
local minima and maxima, 


that is, places where the slope is zero OR
... 


endpoints. 





So to
find the absolute minimum and maximum function values, 


round up
all the suspects and solve the function for each of them. 





We've
been finding places where the slope is zero for a while (F'(X) = 0), 


so that isn't a big deal, 


but what
about the end points? 


How do we
know what the endpoints of a function are? 





There are
a couple of ways we can get to the endpoints. 


The
easiest way, is if the problem gives you endpoints. 


This
would be like when the problem says something like: 


"What's
the maximum value of the function for X values 


between one and a
million." 





The other
way is if the function itself determines the domain. 


That is, the
allowable X values. 


For
example, say you have the function: 











If
we're talking about real numbers only (no imaginary stuff),



then we
can't deal with X values that are negative numbers. 


That
makes zero an endpoint. 


OK, but
what about the other end of that one? 


Actually,
there isn't one. 


X can take positive values all the way to infinity,
and infinity is not a number. 





There
is a special type of notation used to describe values X can have. 


It
lists the endpoints between brackets. 


If
the endpoint is included in the domain (the values X can
have), 


the
notation uses a square bracket on that side. 


If
the endpoint is NOT part of the domain, 


then
the notation uses a round bracket on that side. 


And
infinity always uses a round bracket. 





Examples: 
0
< X £
23 
(0, 23] 
5
£
X £
100 
[5, 100] 
X
³
100 
[100, + ¥) 






Even
if the endpoint is not included in the domain, we still need to
check it. 


If
it turns out that the not included endpoint is the absolute max, 


then
the function DOES NOT have an absolute max. 


If
it turns out that the not included endpoint is the absolute min, 


then
the function DOES NOT have an absolute min. 





That
is because there is no number that is right next to a not included
endpoint. 


It's
one of those cosmic infinity things, 


but
there is no real number that is right next to any other real number, 


and
no point in time that is right next to any other point in time. 


You
can go wacky thinking about that one too much. 





Leaving
the Twilight Zone and getting back to our topic: 








Example: 





Look
at the function F(X) = X
^{2} 








This
function has no limits on the domain (the values X can be), 


it
goes from negative infinity to positive infinity. 


And
as it goes out in either direction, the function value keeps getting
larger. 


So
the function F(X) = X
^{2} has no maximum. 





It
does, however, have a minimum. 


Doesn't
it? 





Finding
it should be really easy. Like review. 


But
let's do it anyway. 





Find
the first derivative: 





F(X)
= X^{2}



F'(X)
= 2X






Set
it equal to zero and solve for X 





F'(X)
= 0 = 2X



X = 0






Now
find F(0) 





F(X)
= X^{2}



F(0)
= (0)^{2}



F(0)
= 0






So
the function has a minimum of 0 


that
happens when X = 0. 





A
function can have more than one piece to the domain. 


The
domain could be: 


(0,3] and [5, +¥)






When
this happens, it's no big deal. 


It
just means that there are more points to check. 


But
no matter how many pieces there are, 


there can only be one absolute
maximum and
one absolute minimum. 





One
way that a function might have more than one piece in the domain 


is
if it has a vertical asymptote. 


When
that happens, 


use the limit as X®
the value that makes the denominator zero as the endpoint. 





Most
of the time, vertical asymptotes make the graph do things like this: 








SUMMARY 


To
find if a function has an absolute maximum or minimum: 





FIRST 


Round
up all the suspects. The suspects are: 



1)
All points where the slope is zero. That is, where F'(x) = 0 



2)
All endpoints. These might include: 




a) endpoints
stated in the problem 




b) vertical
asymptotes 




c) the edge of
"X is undefined" areas 




d) ±
¥ 





SECOND 


Find
the function value at all the suspect points. 





THIRD 


Chose
the maximum and minimum function values 


from the values computed in
the second step. 


If
the X values that created them are numbers that are part of the
domain, 


that
is the maximum or minimum value of the function. 


If
those points are not included, or if they are at ±
¥, 


then
the function does not have that extrema. 





copyright 2005 Bruce Kirkpatrick 
