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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
].
1.
Verify that the function is continuous on the interval [
a,b
].
2.
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.
3.
Evaluate the function at the critical points found in step 1 and the end points.
4.
Identify the absolute extrema.
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.
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This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.
 Fall '08
 sc
 Derivative

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