Finding Absolute Extrema
It�s now time to see our first major application of
derivatives in this chapter. Given a continuous function, f(x), on an
interval [a,b] we want to determine the absolute extrema of the
function. To do this we will need many of the ideas that we looked at in the
previous section.
First, since we have an interval and we are assuming that
the function is continuous the
Extreme Value Theorem tells us that we can in fact do this. This is a good
thing of course. We don�t want to be trying to find something that may not
exist.
Next, we saw in the previous section that absolute extrema
can occur at endpoints or at relative extrema. Also, from
Fermat�s Theorem we know that the list of critical points is also a list of
all possible relative extrema. So the endpoints along with the list of all
critical points will in fact be a list of all possible absolute extrema.
Now we just need to recall that the absolute extrema are
nothing more than the largest and smallest values that a function will take so
all that we really need to do is get a list of possible absolute extrema, plug
these points into our function and then identify the largest and smallest
values.
Here is the procedure for finding absolute extrema.
Finding Absolute Extrema of f(x) on [a,b].
- Verify that the function is continuous on the
interval [a,b].
- Find all critical points of f(x) that
are in the interval [a,b]. This makes sense if you think
about it. Since we are only interested in what the function is
doing in this interval we don�t care about critical points that fall
outside the interval.
- Evaluate the function at the critical points
found in step 1 and the end points.
- Identify the absolute extrema.
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There really isn�t a whole lot to this procedure. We called the first step in
the process step 0, mostly because all of the functions that we�re going to look
at here are going to be continuous, but it is something that we do need to be
careful with. This process will only work if we have a function that is
continuous on the given interval. The most labor intensive step of this process
is the second step (step 1) where we find the critical points. It is also
important to note that all we want are the critical points that are in the
interval.
Let�s do some examples.
Example
Determine the absolute extrema for the following function and interval.
Solution
All we really need to do here is follow the procedure given
above. So, first notice that this is a polynomial and so in continuous
everywhere and in particular is then continuous on the given interval.
Now, we need to get the derivative so that we can find the
critical points of the function.
It looks like we�ll have two critical points,
 and
 .
Note that we actually want something more than just the critical points. We
only want the critical points of the function that lie in the interval in
question. Both of these do fall in the interval as so we will use both of
them. That may seem like a silly thing to mention at this point, but it is
often forgotten, usually when it becomes important, and so we will mention it at
every opportunity to make it�s not forgotten.
Now we evaluate the function at the critical points and the
end points of the interval.
Absolute extrema are the largest and smallest the function
will ever be and these four points represent the only places in the interval
where the absolute extrema can occur. So, from this list we see that the
absolute maximum of g(t) is 24 and it occurs at
 (a
critical point) and the absolute minimum of g(t) is -28 which occurs at
 (an
endpoint).
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