Calculus Absolute Maxima and Minima
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Ups and Downs
Absolute Maxima and Minima

 

 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 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) = X2

 F'(X) = 2X

 
 Set it equal to zero and solve for X
 

 F'(X) = 0 = 2X

  X = 0

 
 Now find F(0)
 

 F(X) = X2

 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

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